1) Do I need an employment attorney or startup lawyer - or are they the same thing?
2) Can I legally do this? What sort of contract do I need?
3) What amount of equity would be fair for me? Vesting cliff?
I am involved with Swedish company in a similar sounding arrangement. While not Indian I think the overall experience will be similar.
The first step should be to find a tax expert whole is familiar with taxation issues regarding India. Most countries have tax treaties with the US. This a key step to understanding what the best arrangement can be. As one person suggested, and as happened in my case, it was best to setup a company in the US. Because of the foreign ownership component it had to a C corporation. We had considered the other obvious options: being a consultant, being an employee, being a partial owner. All of the other options were ruled out because of the tax burden. The unfortunate aspect of this is that unless you are dealing with a large law firm international taxation is a relatively specialized field and the tax lawyer and the lawyer to help with the business contracts will not be the same person. On the positive side India should be much easier than Sweden due to the amount of business between the countries.
Once the taxation strategy is figured out then a business lawyer that is familiar with startups would be recommended. I would suggest that setting up the US business takes place before the contract with the Indian company is signed. However, I would suggest having a clear outline of the intent and deliverables of the contract is important before you finalize the US company. I was lucky in the fact that my business lawyer helped with both the startup and the contracts.
A note on the contracts: as many people have said it will be nearly impossible to enforce a legal contract merely from the cost of a legal proceeding in a foreign court or in a US court with foreign litigants. As others have mentioned the contract is more of a roadmap of intentions, agreements, and deliverables. I would highly encourage an old school face to face meeting prior to signing the final paperwork.
While I have personally worked through question 3 I believe that this is different for each situation and what each party is bringing to the table. I am sure that others have more experience to offer in this arena than I.