Is there a limit for devices where I can download my purchased apps? I'm thinking about renting an iPad for use. And I would like to know if I can put my password to download my apps every time I rent an iPad, or is there a limit?
You can download any app and other media purchased with your apple ID to a new device - as long as you authorize your new device with that apple ID.
This is what the official Apple documentation says:

Your Apple ID can have up to 10 devices and computers (combined)
associated with it. Each computer must also be authorized using the
same Apple ID. Once a device or computer is associated with your Apple
ID, you cannot associate that device or computer with another Apple ID
for 90 days. You can view which devices or computers are currently
associated, remove unused devices or computers, and see how long
before they can be associated with a different Apple ID from the
Account Information page in iTunes on your computer.
Last Modified: November 14, 2011