HTHThreee

It's engineered, not natural/random.

Life is isotropic, so it remains symmetric vertically. It travels horizontally but I don't know which direction because it's so unfathomably large you would need a supercomputer to run at least a cycle of it.

Smoke refers to fairly long lived debris that eventually dies out, the term for one that leaves permanent debris is a puffer.

No, if it does it's called a puffer.

OK, I didn't know if it was a "true" replicator (what even does "true" mean) so I excluded it. And, running a true replicator from other rule in 0E0P takes millenia, so that leaves DOGun SaGaQR (specifically the QR configuration). Sorry.

For software, use Golly, it has HashLife (pretty fast algorithm), can open many types of files, and supports scripting.

I think of reality as of GOL but 3D, with more states other than 0 and 1, and conservative (follows conservation laws, no relation with any politics).

yeah, spaceships are pretty common (fun fact: 2 spaceships were found on the same day including this one in GOL), also in CA it's called a "spaceship". However until now there was no 1D one (1D is also called 1-cell-thick or linear). Also, you are actually wrong, it is actually much easier to move than to self-synthesize and then remain alive like a replicator. There has been no true replicator found yet in Life as far as I know (arguably, linear propagator may be a replicator) but like millions of spaceships have been found.

conwaylife.com isn't just for GOL, although it is the main subject, there is an Other Cellular Automata forum. also it's not really a fansite, it's a bit more academic than what would be considered a "fansite".