I have downloaded Photoshop and Bridge to my desktop. Do I have to pay again to have it downloaded to my laptop?

no.

one single user subscription allows you to install cc apps on an unlimited number of computers.

the only restriction is that you can be signed-in to, at most, two computers at any one time.  because adobe made it so easy to sign-in/out from any computer, even that's not much of a restriction.

i use cc apps on 4 computers and the inconvenience of signing out/in is minor.  using only 2 computers, you don't ever have to sign-out/in after your initial sign-in on each computer.