Given the universe is expanding — with every galaxy flying away from every other galaxy, why do the Milky Way and Andromeda galaxies find themselves on a collision course?
Expansion is the average behavior. It doesn’t mean that literally every body is always expanding away from every other body all the time.
To use a different example, the average global temperature has been rising for a while. But that doesn’t mean every year is warmer than the previous one. The trend has been positive but there are still year to year fluctuations.
Things like solar systems, galaxies, and galactic clusters are gravitationally bound. That means that gravirty dominates as the major force at this scale.
Imagine throwing a clod of dirt with great force and it breaks up as it flies away. But individual sub-clods stay together due to chemical bonds and these sub-clods spread out away from each other.
Then imagine ignoring the previous example as being not quite analogous.
Because the 2 galaxies are gravitationally bound to each other and that binding is sufficient to overcome “local” expansion.
The further something is away from us, the faster it is receding. Andromeda is sufficiently close that it is moving towards us faster than the distance between us is expanding.
Now I have to worry about our galaxy colliding with another galaxy. You didn’t have to tell me this, you know. Ignorance is bliss.
It’ll be total chaos. I dread the day the stars in the Milky Way and Andromeda start ricocheting off each other like a hard break in billiards. I’m wearing an inner tube in anticipation. 
Or to take a more mundane example, the fact that the universe is expanding does not stop you giving someone a hug…
Here’s a fun article on the topic.
I dunno. The way I’m expanding, maybe not for much longer 
I think you were joking, but no actual stars would actually collide, but just all be moved around through gravitational effects. I once heard this (perhaps here): if the Milky Way galaxy was the Pacific Ocean, and every star was a grain of sand, then the grains of sand in the Pacific would be a mile away from each other. So the chance of an actual collision would be minuscule.
Yeah, it was a joke. The Milky Way colliding with Andromeda will just result in a larger galaxy.
Car A is driving east.
Car B is also driving east.
Car B collides with Car A.
How is this possible?
The solution is left as an exercise for the reader.
Car B is behind Car A and driving faster? Or, do you mean a head-on collision?
No stars will collide (probably), but planets and stars can be and have been hurled right out a galaxy by the interactions of a ‘collision’. For that matter, our galaxy is already full of rogue planets that were flung out of the gas and dust discs that created them.
I’d miss our sun. It’s been very good to us.
Here is another article with some further developments on the Andromeda/Milky Way mashup.
With the understanding that the Sun will likely have already swallowed up the Earth by the time the galaxies merged (or perhaps right around the same time), would it be possible for our entire solar system (or any other solar system) to be flung out into space and remain intact? IOW, could a solar system remain viable out in space outside of a galaxy, and could the life that potentially exists within such a solar system remain unharmed throughout the process?
If so, I feel such a system would actually safer outside of a galaxy, where there’s much less chance of getting too close to a supernova, black hole, gamma ray burst, etc.
Was the reader driving? That might explain it, you should not read when driving.
I really don’t think the solar system would survive intact if the sun were hurled out of thr galaxy by a close interaction with another star.
Our galaxy is already colliding with Andromeda. Both galaxies have a very large and diffuse halo of stars and dust around them, and our halos are actually touching already. But yeah, by the time any of that could affect us we’ll be long gone.
It’d be quite extraordinary if the Solar System didn’t survive intact with the Sun being hurled out of the Galaxy. And while a planet stripped from its sun is in big trouble, a solar system doesn’t really have much need for the rest of its galaxy.
Ungrateful solar systems. Took all those great heavier elements from old supernovas and what-not to grow up and then take off without so much as a “thank you”.
One part of a galactic encounters not mentioned as often as it should is the effects on stars in one galaxy caused by the dumb luck of the core black hole of the other galaxy passing at all nearby. It would be a Very Bad Day for a solar system in such an encounter even if no actual “collision” takes place.
While the most likely scenario is the core passing through ~ the narrower part of the other galaxy, a full lengthwise pass would be possible and affect a considerable number of systems. And keep in mind that during a merger such passes happen over and over again.