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If every star and planetary system and blackhole—all mass (not sure whether to include dark matter or not) except what directly makes up our solar system—in the Universe except the Sun were wiped from existence (not explosively destroyed) would the existence of life on Earth be immediately (~within a few years of us noticing they'd disappeared) threatened?

We wouldn't even notice for several years.

The closest star to us (aside from the Sun) is Alpha Centauri, which is just over 4 light-years away. That means that whenever we look at Alpha Centauri from earth, we are seeing light that left the star over 4 years ago. If Alpha Centauri were extinguished today, we wouldn't even realize it until 4 years from now! For most stars, it will take decades, centuries, or longer before we see them wink out of existence - whatever effect that might have, it will definitely not be "immediate".

No. Life on Earth and our solar system in general would not be harmed by this sudden universal destruction.

Everything outside of the Solar system affects us via electromagnetic radiation, gravity, and "matter transfer".

The EM radiation flux is too weak to really do much, other than marvel at through telescopes. With the rest of the universe gone we'd actually be safe from the potential of a nearby Gamma Ray Burst.

The gravitational influence of our stellar neighborhood is far too weak to majorly change anything like planetary orbits. Same goes for the galactic center we orbit around. So probably no kicking asteroids or comets on a collision course with us.

The final chance for an influence is through "matter exchange". There's no longer the chance for big rocks to come sailing through our solar system from beyond, so again we're safer here. No more worrying about rogue black holes or ejected exo-planets disrupting local orbits.

We do lose Cosmic Rays in this ordeal, though. Cosmic rays produce most of the Carbon 14 we could use for radio-carbon dating, but we've already messed that balance up through weapons testing.

I don't know enough about biochemistry to attest to the importance of the isotopes produced by cosmic rays, but at a casual glace and google they don't seem to be terribly impactful or necessary.

So without cosmic rays our background radiation levels would decrease and electronics would become slightly more reliable. Once again we are made even safer.

Ultimately we will perish as a species when the sun goes through its stellar life-cycle and the solar system (now entire universe) cools... but that'll probably happen anyways on a longer time scale if the Heat Death of the Universe holds true.

Nothing would happen. Solar system is self-sufficient enough. It would be only little influenced if rest of universe somehow ceased to exist.

That said, anything that would preserve our little bubble intact and simultaneously destroyed everything else, is almost certainly unimaginable to science.

There would be basically no effect.

First off, even gravitation is governed by the speed of light, so we would not notice for thousands of years (well, we'd notice Alpha Centauri after about 4 years). By the currently understood laws of physics, no information about a remote object can travel faster than the speed of light. We theoretically will see some hilariously small quantum fluctuation differences, but we wouldn't actually be able to prove anything from them until information traveling at the speed of light catches up.

However, this can be taken in another way. Since you're looking at gravitation, we can compute the effects. Distance is a major player in this equation, so lets pick the closest star, Alpha Centauri. It is 4.3 light years away. that's $4.132cdot 10^16 text m$ away. Its mass is roughly 0.123 solar masses, or $2.446cdot 10^29 textkg$. Throwing them into $a_grav=fracGMr^2$ we get an acceleration of $9.5cdot10^-15fractext mtext s^2$(G is $6.674cdot10−11 fractextm^3kgcdot s^2$, if you want to run those numbers). So while the earth is pulling on us at $9.8fractext mtext s^2$, A. centauri is pulling with a whopping $0.0000000000000095fractext mtext s^2$

Now note the r-squared term. That's why the gravity of Earth is such a big deal and A. centauri, as big as it is, isn't a big deal. The gravitational effects of far away objects is pretty negligible. If there was something that kept emitting light where A. centauri was, we wouldn't even detect it.

In the long run, these small things would matter. If the galaxy vanished, we would stop orbiting the galaxy. Our orbit around the center of the galaxy has a period of about 200 million years, so you'd start to notice the loss of that mass after a million or so years.

The bigger issue would be the worldbuilding problem. Something just broke all of our known laws of physics, and did it in a blink of an eye. What was that thing? How does it work?

And is it still hungry?

These are important questions for the denizens of the last surviving solar system in the universe.

It would take time, but yes life would cease as we would have fallen out of the the fabled "Goldilocks Zone" (Wikipedia link here). That said, like Nuclear Wang said it would take time. The other thing to take into account gravitational mass that happens when a star dies - generally it becomes a Blackhole. So, this would also throw our solar system into a state of disarray and things would change as orbits changed/decayed. This could speed up the process or not --- depending a huge array of variables which go far beyond my understanding (where is Neil Degrasse Tyson when you need him!)