I am about to rent an apartment. The owner wants rent plus deposit in cash, and the lease and keys will be provided later. I am concerned about giving so much money without some assurance that I will have a legal right to occupy the apartment. Would it be appropriate for me to insist on a signed lease at the time of payment?

Yes, do not just hand over cash to anyone without them giving you a receipt first. I would recommend that you get a money order or write a check. And, every lease that I have ever signed I have had to pay first and last months rent and when I paid them, I signed the lease and received the keys all at one time.

Whatever you do, make sure you have a papertrail in case they try to rip you off, because you can't take them to court without any proof of every giving any money.

This sounds pretty fishy to me of the owner. I have NEVER heard of that.