Chapotraphouse

Interstellar space is so empty that it's not going to experience significant erosion from micrometeorites even over geological significant lengths of time. The interface points between dissimilar materials can be a source of corrosion, so certain portions of the structure will degrade in the relative short term. I don't know exactly what techniques and materials are used to attach the various parts of the structure, so I could imagine that galvanic corrosion would eventually cause the various booms to collapse and it to turn into a jumble of equipment that is just kind of flying through space. But there's no particular reason that it won't remain a recognizable artificial object billions of years from now - if there's no mechanism for an object to degrade, it just kind of continues existing. Take for example fossils such stromatolites, which are billions of years old, but still recognizable for what they are when examined by intelligent beings.

Will the golden record last a million years? Certainly. A billion? Probably, but there might be some process or environmental factor we don't yet know about that will cause it to degrade. A trillion? That's an awful long time, it seems like a bad bet to make... but it's not out of the realm of possibility.

Hitting an object straight on is certainly unlikely, I'm not so sure about getting caught in a gravity well and becoming a satellite. I imagine the escape velocity of the milky way on the whole would have something to say about it actually leaving, too.

Over millions to tens of millions of years they'll be gradually eroded by cosmic ray spallation. They don't have the velocity to escape the galaxy so they'll kind of continue as they are, free drifting through space like everything else.

Newton's first law, they will move until something stops them. Like you said, space is very empty and they likely won't ever collide with asteroids or anything. If they get close enough to a planet, which is unlikely, they could fall into orbit and then that orbit will decay until they crash.

To even exit our solar system is a task. They still haven't left and it's been decades. Every now and then you'll hear news about how they're leaving or have left, but they're still technically in our system and will be for some time. Humans will probably be dead before they even leave the influence of our solar system.

Once they reach interstellar (between solar systems) space, they will just keep going, probably never reaching another galaxy. Voyager has like 721 kilograms of mass. If you took the average atomic weight of the materials its made of, and supposed it lost some amount of atoms per day due to microscopic collisions, you can ballpark the time until it turns into dust. Without doing the math, I bet it would be just a hunk of metal before even getting one lightyear away from our solar system.

This question caused me do some tangential navel gazing. The video mentions an escape velocity of 550km/s to leave the Milky way. According to some random stack exchange reply relative to Earth you will only need 317 km/s ∆v (because Earth is already orbiting around the galaxy at around 220 km/s).

I wonder how big such a rocket would be.

They are then unlikely to be moving fast enough to actually cross intergalactic distances quickly enough for the expansion of space to not outpace the distance covered, leaving them in this void forever.

Well, fuck, I hadn't ever thought of that.