Which leads me to think about the people who say that if they found they were living in a simulation, they'd try to get out. Unless the simulation is very similar to the substrate, would it be possible to get out while remaining yourself in any sense?

Sure. Escape into another simulation.

More seriously, obviously it's not guaranteed that an organism in a simulation can just create a copy in the outside world. How would a Game of Life organism, made out of glider guns and flashers and whatnot, made an atom-based form of itself?

What it could do is create something isomorphic. Whether this is possible is pretty much the same question as whether humans can make uploads. (Which is the inverse, actually - going from 'reality' to 'simulation'.)