I could not get Elbrus and Cygwin to plate solve so I installed Astrometry.net and things are working fine. Can I safely remove the Elbrus and Cygwin files or does Astrometry.net somehow need them?
The local astrometry.net server (ansvr) installs and uses a copy of cygwin. Elbrus does not use cygwin at all.
If you ended up with an extra cygwin somehow then sure, you can remove it. The ansvr installer installs cygwin in C:\Users\YOUR_USERNAME\AppData\Local\cygwin_ansvr so you must not delete that one.