Do my downloads continue while my laptop is in sleep mode?

Yes, they will stop and most likely be canceled altogether.

When I upgraded to Mt. Lion from OS 7, I had to take the Mac Air to a dealer because the DSL speed I use was too slow to download and install the whole thing. It took 4–5 hours even at the dealer’s.

Your downloads will stop while your laptop is asleep, but will resume when you switch on in the morning. Depending on your settings, of course.

Sorry I only know how real computers work. I never bothered with a Mac.
On a Windows 8 based computer, in sleep mode file transfers stop where they were and depending on the transfer protocol either resume when the PC makes up or is cancelled by the server.