Is there typically a signed lease or other contract when one rents a furnished apartment for a couple of months? I'm hesitant to give a deposit without a written agreement.

You bring up a good point. I booked an apartment in Sevilla and sent a 400 euro deposit without an agreement. When we arrived in August, we signed the lease. Probably naive, but so far, so good. I felt a little more secure because a friend from Seville checked out the apt., met the owner of the agency and knew him as a business man about town. Still pretty naive. I think it's a good idea to ask if you can sign the lease before sending the deposit. A reputable owner should have no problem with that. Besides, they are probably tickled pink to have the apt. rented out for a few months straight. Good luck!