You can hand in other lessons while waiting – yes. I had one course that I completed and submitted all the homework before taking the 2nd “midterm”, but that was when most of their courses were web-enabled (I took all my accounting courses through LSU, and only one was “online”, the rest were “web-enabled”), and I think my proctor was on vacation when the exam arrived so I had to wait a bit to take it, or something like that.
I think officially, as soon as you've submitted the last required pre-midterm homework, the midterm is sent to your proctor. Practically speaking, it may take a couple days for that to process. It's sent through the postal service/”snail mail”, so will take a couple days to get to your proctor. So, probably count on about a week from when you submit the last pre-midterm homework. If your proctor is pretty flexible, you'd probably be safe to schedule it for a week out, but if re-scheduling will be a problem, then you might want to wait till s/he receives it. This may be quicker now that most of their classes are online and they have less paperwork to keep up with, or slower since they don't do it as much.
After you take your exam, your proctor mails it back to LSU, who sends it on to the professor, who grades it. So, the grading process takes a bit. I'd see how long it's taking your professor to grade your homework, then add 4 business days to that to get an estimate of how long it will take to get the exam graded. You don't get back a graded copy (or at least didn't used to), just have a grade show up, so don't worry about waiting to see what you did well on or poorly on.
.