I understand the whole process of how equity should be divided, but what is the actual technical process of distributing equity? Is it a contract? How does someone sell their equity? If I want to give someone 30% equity, should I just write it on the back of a napkin?
EVERYTHING is a contract. Yes, it is - the intiial distribution is part of the contract forming the corporation.
The rest - your legal advisor will tell you. Depends on jurisdiction - shall we assume you are in the US because you dont care to say anything about your country?
Again, depends ON YOUR COUNTRY - and the legal form. Sometimes you need to hand over physical shares, sometimes you need to make a eneral assembly of all shareholders in front of a notary.