What paperwork do we need signed by our partner that is leaving the company, and giving up his equity and rights to all projects he worked on?

it depends on what your partnership agreement said but ultimately you should be able to create or execute on your buy sell given the terms of your partnership agreement. He should also forfeit any stock shares issued. If none of that has been done if the partnership was never inked properly, then it would just take an attorney to draft the dissolution paperwork and take him off any shareholder docs, etc.

You need a shareholders agreement which effectively acts as your insurance policy for these kind of eventualities. A lawyer will be able to draft an appropriate agreement according to the legal entity your business operates as. Before you start any venture you should have one drafted. Having it in black and white will allow you to sleep at night ;)

We had a similar situation with a co-founder in a previous venture of mine. It just wasn't working out, as he was unable to make the jump to a full-time commitment and that's what his position would have required.

The entire separation process diverted a lot of time and resources away from the company and our pursuit of the mission. That said, we did end up working with an attorney to tie up the separation as neatly and cleanly as possible.

My suggestion would be to find a qualified attorney to draft the agreement. That way you won't be spending any additional time and energy trying to put the situation behind you, since you'll have made the investment to handle it the right way, the first time. Also, if there is any intellectual property (IP) involved, make sure the IP rights are assigned from the departing partner to the company.

Happy to discuss further if that would be helpful and good luck!

Chris