You should try to get some sort of contract, signed by the client. You can include the arrangements you're asking about.
- But how do you get paid?
Directly into your bankaccount is safest, I guess. But that's up to you.
- How does an end client typically pay an agency?
I've no idea.
- Is it okay to ask the client to pay within 30 days after the completion of the project, as if I am getting paid by an agency?
Of course it is, especially if you include this in the contract.
- How do these translation agencies get paid by their clients? Any insights?
Same way you get paid, of course. Just a bit more
- Oh, would you worry about "legal" issues? I think we are typcially sheltered by legal terms established by the agencies.
That's why you need to get a signed contract before you start the work. By the way, working with an agency doesn't make that much of a difference. What do you do if they don't pay you?