This is my experience (not legal advice, ymmv, etc, etc.). I have done this when I had an H1-B. I was classified as remote. There are two implications:

1. Prepare for USCIS to visit your home. They want to see if you're really working ;). I had someone come and showed her around. Make sure you have a designated "office space" in the house. This was in 2012 or 2013 so I'm not sure if they still do it.

2. Obviously, if you're applying for your green card, then recruitment happens country-wide as opposed to just being in your local zone. At least this is what my lawyers said (again around 2012-2013 :) ).

Yes. It just needs to be disclosed as part of the H-1B process.

I did this on H1-B. Your employer may or may not need to change the "statistical zone" where you are working if you move around, because salaries need to match it. A work location change is an easy H1-B amendment but you need to file it, and it may also reset your greencard application on some cases. You should really talk to your employer about it, they will need a lawyer.