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I would like to understand in which cases the word distance is a mass noun and in which it is not. Indeed, distance is something that one can measure, so for me it looks like it should be a mass noun. However one says.

Please keep your dog at a distance.

Question 1. What is the rule here?

Question 2. Are there some similar situations in which a "naturally mass noun" becomes countable?

The word distance is meaningful in both an uncountable sense and a countable sense. So, the type of determiner (or lack of any determiner) that you precede it with, or whether you use the plural, indicates how you want the listener to understand it.

First, here is uncountable distance:

Let's put some distance between us and Houston. [That is, let's drive away from Houston.]

The force of gravity is proportional to both mass and distance.

It wouldn't make sense to put "distances" in either of those sentences.

The countable sense of distance is any specific distance, like 2.1 meters, 20 miles, or 3 inches. You can count these:

This star map shows the distances from Earth to 2,000 different stars. [The distance from Earth to Alpha Centauri, the distance from Earth to Sirius, the distance from Earth to Polaris, etc.]

My first day on the job as a surveyor, I measured 47 distances.

No one would really say the second one, but it does make sense.

Once you understand countable distance, then you can understand common expressions like at a distance and a distance.

I watched the battlefield from a safe distance. [The distance between me and the battlefield was great enough that the bullets couldn't reach me.]

The arc of a baseball in flight isn't caused by a force that Earth exerts on the ball at a distance, as Newton's theory held. Instead, the ball is responding to the gravitational field immediately around it. [Source: Spooky Action At a Distance (2015). Action at a distance is direct causal influence by one object on another object that is spatially separated from it. In other words, there is a non-zero distance between the objects.]

Please keep your dog at a distance. [Please keep your dog far enough away from me that I don't need to fear getting bitten.]

These countable uses of distance don't specify a distance numerically. They refer inexactly to a distance: whatever is far enough for the purpose under discussion, even if only implicitly, as in the sentence about the dog. However, it would be strange to put "distance" into the plural in any of these last three sentences.

Very interesting question. I don't think the inclusion of the indefinite article here is not so much making distance a countable noun – although I can understand how a learner might parse that sentence and draw that conclusion.

Instead, the phrase at a distance is a standard expression – one that gets its own entry in some dictionaries.

For example, from Macmillan:

at/from a distance (phrase)

at/from a place that is not close
I’ve only ever seen him from a distance.
Tim followed him at a distance (=keeping a long way behind him).

and from Collins:

at/from a distance (phrase)

If you are at a distance from something, or if you see it or remember it from a distance, you are a long way away from it in space or time.
The only way I can cope with my mother is at a distance.

But I think your question is even more interesting because distance (like many other nouns) can be used in both a countable and uncountable sense. If you look through the definitions in Macmillan, you'll see the UNCOUNTABLE label used six times, but COUNTABLE label used twice.

This is a case where you can't really judge the noun's countability based on the articles around it.