Well, first, if you are accepting equity in an existing business then you will have to pay income taxes on the value of that equity. If the business isn't worth anything then fine. If it has income or a value then you will have to report that and pay some taxes.

Equity pays out during the year or during the life of the business in the form of distributions. These are untaxed, at the time, payments out of the profit to the shareholders of the company. At the end of the year, each of the shareholders will get a form that tells them what their distributions amounted to and they have to pay taxes on that. Not all companies do distributions and there are some rules about what percentage of your income can come from distributions versus salary from the same company.

Equity pays out when a business is sold, yes. Much like above, the shareholders split the net based on their equity share after all other debts are settled.

I'm not sure what you mean by a partner leaving. A working partner could stop working, stop collecting a salary and still retain their shares. They could keep working and yet sell their shares to someone else. They could both quit and sell their shares. They two don't necessarily have to be related. In any case though, if a person sells their shares then they get paid based on the value of those shares. Privately owned companies should have rules about who can sell, for how much, to whom, etc. since equity is often related to decision making. For public companies there are some special classes of stocks but I don't think you are asking about any of those.

Finally, there is the intangible benefit of decision making just mentioned. As an equity owner one can expect to exert some amount of influence based upon the size of their equity. In a very small organization one person owning 51% or more might run it. In a very large organization, a group owning as little as 5% might exert much more control than anyone else. In any case, it is traditional that shareholders have some say in how the company is run or by whom.