Yes. You can actually measure this just by the mass difference of a mountain here on Earth, with a sufficiently accurate pair of clocks.

http://www.leapsecond.com/great2016a/index.htm

Yes, it does (from the point of view of observers outside the gravity well), and yes they could. They could also accelerate themselves to relativistic speeds to accomplish the same feat.

I think they’d just need to travel slower through space to experience more time relative to the other civilization. Then that civilization could take more that time to develop a civilization-ending laser or whatever.

Then for gravity, the less gravity that civilization experiences the slower time moves for them relative to their competitor.

So I think the goal would be to send star matter to the other civilization so their time slows down relatively speaking.

I’m no expert, though. Happy to take corrections.

Would that not achieve the opposite effect?

If interstaller thought us anything, the closer you are to a black hole the slower time gets. When they got close to the black hole they only spent a few hours there but lost years compared to our frame of reference.