I think so. Search for certifications by company name under PERM at https://icert.doleta.gov/index.cfm?event=ehLCJRExternal.dspLCRLanding . Under section 8, you can see current applicants status.
Yes. They do.
You need to be level 5 (/Sde2), to start GC process.
I would not do this if you plan to travel in the next 3 years. H1B allows for dual intent (immigrant and non-immigrant). F1 does not.
Lmao.. at least google some basics..