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A hopefully not-too-long not-too-short background

The bicameral legislature of the USA occurred as an early compromise among its founding fathers, between those who believed in individual ownership of governance [where comparatively more power ends up invested in urban regions] and those who believed in states’ ownership of governance [comparatively more in rural regions]; the compromise was just “nothing becomes a law unless both groups can agree on it.” The Senate gives each state two votes; the House presently attempts to give each subset of N people one vote for some N by using a census to divide each state into districts where each district, hopefully having around N people, runs a first-past-the-post election.

The danger of gerrymandering by those who draw the district lines is one reason to avoid such a system, but since the Senate seems to have a similar 50% Democrat/50% Republican deadlock it is probably not the cause of the dysfunctional US government. Indeed it rather seems likely that this process of splitting a country into segments that each run first-past-the-post elections leads inexorably to dysfunction and then civil war, and one can even view the US Civil War as having been ultimately caused by this by looking at the partisan divisions beforehand. First-past-the-post schemes contain a well-known spoiler effect where voting for a third party who better represents your interests can ultimately cause your interests to be more-poorly-represented; in this context it seems that there is a mathematical fixed point where a country contains two fixed polities who each vote passionately for one candidate, not because they support that candidate, but because the other polity terrifies them. (“If they didn't vote for a lizard, the wrong lizard might get in.”) Viewed this way, while slavery was the casus belli of the US Civil War, one might reasonably assume that if some sort of agreement had been reached about slavery, the US would have still had a civil war, just some years later over some completely different issue that sparked the powder keg.

Changing the House

Unless the USA wanted to revisit the earlier compromise, the states would still occupy one chamber and it would be hard to do elections for that chamber other than first-past-the-post (although I can't see anything in the 17^th Amendment to the US Constitution that would block instant-runoff or approval voting or so).

However several nations seem to be more robust against this sort of problem through party-list proportional representation where the people vote on parties rather than individual officials, and those parties receive a proportion of the final elected officials.

The US Constitution is much more lax about the composition of the House, stating only that:

The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature.

[...age, citizenship, residence...]

Representatives and direct taxes shall be apportioned among the several states which may be included within this union, according to their respective numbers, which shall be determined by adding to the whole number of free persons, including those bound to service for a term of years, and excluding Indians not taxed, three fifths of all other Persons. [...] The number of Representatives shall not exceed one for every thirty thousand, but each state shall have at least one Representative [...]

This last paragraph was modified in detail but not in spirit by the 14^th Amendment.

Most proportionate allocation systems can be modified to allocate simultaneously over both states and parties. Am I correct in understanding that as long as a proportionate representation system also distributed representatives in the House proportionately among the various states, that no constitutional amendment would be needed to make the switch?

Am I correct in understanding that as long as a proportionate representation system also distributed representatives in the House proportionately among the various states, that no constitutional amendment would be needed to make the switch?

No.

is the fact that states choose their own representatives something which is part of some law passed by Congress some long time ago?

No, the fact that states choose their own representatives is right in the part of the Constitution that you quoted.

The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature.

As a resident of Massachusetts, I do not meet the qualifications requisite to vote in New Mexico’s legislature. I am not an elector in that state. I cannot elect that state’s representatives.

The broader point that you are missing is that the Constitution is a document that describes a union between states that are sovereign in-and-of themselves. This is the fundamental assumption that underlies the entire constitutional order. Allocating seats proportionally in the way you describe across the entire nation, along with many other ways of conducting national votes that are used in non-US countries, would override that assumption.

Joe's answer is great if you're talking only about a single nationwide election. If you are primarily concerned with gerrymandering, however, that can be eliminated by having individual states move to some form of proportional representation. That is not without significant challenges but it is a much lower bar than moving the entire country to proportional representation.

The first issue you'd have is that federal law (beginning with the 1842 Apportionment Act and going through the 1967 PL-196) requires each house district to be single member. That is just a federal law, however, so it could be changed. There have been questions about whether the federal government can constitutionally set this sort of requirement so you could potentially challenge the constitutionality of the law rather than getting it repealed if you wanted.

Next, you'd have an issue that the Supreme Court has traditionally looked unfavorably at multi-member districts when it comes to drawing boundaries for state legislatures. There is no reason to believe that the SCOTUS wouldn't have similar concerns about multi-member House districts. There are arguments that multi-member districts will unconstitutionally dilute the power of minority voters in violation of the Fourteenth Amendment and the Voting Rights Act. If you have a state with 10 districts and a reasonably geographically compact minority population, it would be difficult to draw legal districts that wouldn't ensure that the minority population got to elect a representative of their choice. In a multi-member district, however, they could easily be thwarted by the more numerous majority (potentially depending on where their preferred candidate is on the party's list). You'd need to convince the SCOTUS when the case inevitably came before them that whatever system you'd move to would be equally protective of the rights of voters to what we have today. My wild guess is that it's unlikely but not impossible that the SCOTUS would allow a state to move to proportional representation-- maybe a 1 in 4 chance that it would be upheld.

Currently, there is a federal law against multi-member districts. This federal law was passed to prevent states from having one actual district where the same people voted on multiple representatives. However, as written, it bans all multi-member districts, including any version of proportionality.

That law could be repealed or its restriction of proportional districts could be overturned by the courts (conceivably). I'm not convinced at this time that there is a strong argument for overturning it at the judicial level. The Supreme Court has seen several similar arguments and refused to overturn legislation. For example, most states have laws requiring districts to be balanced by overall population. A court case attempted to make the requirement by citizens or eligible voters instead, but the courts refused to mandate that.

If the law were repealed and replaced with a law mandating proportionality, this might be constitutional. Certainly the federal government has an interest in making sure that elections are fair. One might even make a fourteenth amendment argument. Proportionality would help protect minority rights in states where minorities are scattered throughout the state. They could then aggregate together to pick their preferred candidates.

If the law were simply repealed and replaced with a law prohibiting one person having a vote for more than one representative, that would allow states to move to proportional systems without requiring it (they could continue to use first-past-the-post/plurality in geographic districts).

The original law was created because majority white states were mandating statewide districts in which minorities (generally blacks) were not competitive. However, even without that, it took the Voting Rights Act to mandate minority representation. A proportional system like Single Transferable Vote would shift that power out of the hands of the legislative majority and into the hands of voters. Voters themselves would choose how they aggregate. Minority voters could choose to vote for the minority candidate or they could choose to vote for a candidate that shared their beliefs regardless of race.

From past behavior, we can guess that many will choose to vote for a minority candidate. And of course, some non-minorities may also vote for minority candidates.

Historically minorities had lower voter participation. But during the Barack Obama elections, blacks actually voted at a higher rate than whites. Presumably that would continue if blacks were always able to vote for a candidate of their preference. The same thing might well happen for other minorities, although we don't have as clear a demonstration as with blacks and Obama.

Other proportional systems aren't as strong as STV. For example, in many party list systems, the party chooses the people in the lists. So in a party list system, they might require additional rules to make sure there is minority participation. The German system has the same kind of problem. So it would also need additional rules requiring minority participation to avoid a fourteenth amendment challenge.

This is part of why I prefer STV to these other systems. It allows people to make individual choices, not just choices of party. It is thus more compatible with the fourteenth amendment in its basic form than other proportional methods.

Within states, a proportional system is constitutional although it may require tweaks to fit into the fourteenth amendment.

It is more questionable between states. The basic problem is that the census is mandated as the method by which votes will be apportioned. So to change that to a proportional system would require declaring that proportional system to be a census. That might not fly, as a voting system does not require universal participation where a census does. So to change apportionment might require a constitutional amendment. It really depends how willing the court is to bend the meaning of census. They did bend the meaning of legislature similarly in a previous case. So it is possible. But it's not clear if there is the same support for doing so with the census.

Another challenge is that both Democrats and Republican partisans are against proportional systems. Because statewide proportional systems would tend to allow third parties to be more competitive with at least some seats. This would weaken both of the traditional parties. So partisans tend to prefer systems that would advantage their party. For the Republicans, this is the status quo. For Democrats, they want systems like the efficiency gap, which advantage them at the expense of Republicans while continuing to wall out third party representation.