Of course it has an effect on the environment. Everything does. Tim Dodd did a nice analysis: How much do rockets pollute?
TL;DR: Falcon 9 is better than most rockets, as it doesn't use SRBs, and the first stage and fairings can be reused many times. Starship will be better still, as it uses methane instead of RP-1 (reducing soot), and the entire vehicle is designed with reuse in mind.
Once burnt, yes, safe greenhouse gas. At least, no worse than any other combustion products. I mean, it would be nice if hydrogen wasn't such a pain, but we have to work with the universe we are given.
Methane leaks need to be contained to the amount practical, but they are doing that.
If by "left in space" you mean the payload, then mostly Starlink satellites. A considerable number of other people's satellites as well. Those stick around until the end of their service life, then they re-enter the atmosphere and burn up.
If you're asking if any part of the rocket gets left up there, then the answer is no
Right. It’s more like the molecules come apart from one another but still exist. Kinda like when bread turns into toast in the toaster: matter transforming as energy flows though it.
Dont worry, according to elonmusk.today, it's been 679 days since he anounced they would make rocket fuel from carbon in the air. So, that program is bound to produce results, right? riiiiight?