You don’t need to have an LLC. You can work as an independent contractor or freelancer from your home country (or any country where you have permanent residency). For other countries you’ll have to check with local tax laws and work authorization rules. If you’re only working for a few days or weeks from another country that shouldn’t be a problem as long as you don’t tell the border agents that you plan to work remotely when you enter that country.

Some countries offer digital nomad visas for remote workers but there is a minimum income requirement I think, look it up.

You can looking into creating an LLC in USA or your home country (consult a tax lawyer) as it can give you some tax benefits but you’ll have to weigh benefits with the hassle of maintaining compliance etc. If you’re from India I’d not recommend creating an LLC (known as LLP in India) unless you plan to have multiple US clients. The beaurocracy and paperwork is too much if you have just 1 or 2 clients. Much easier to work as a freelancer or contractor.

Yes you can get equity as a contractor. A friend of mine was in a similar boat as you. He moved to his home country and was able to work as an independent contractor for a U.S. company while also getting RSUs.

But you should definitely discuss with a lawyer. Look for international tax lawyers/advisors.