Ever notice that most buildings to not have a 13th floor?
Be sure to locate your 13 access points in a manner where those on different non-overlapping channels, do not hear or interfere with each other. You'll need a channel plan to prevent mutual interference. I strongly suggest you read through this document from Intel on "hot spot" topology and planning before blundering onward:
Not a good idea.
Yep. That's exactly what will happen if you do it that way. You should use a single SSID and different channels. Use non-overlapping channels such as 1, 6, and 11.
Yep. However, be advised that some client radios will stick tenatiously to one access point before finally deciding to let go and scan for a better connection. This arrangement has a problem where one is not being able to select a specific access point or even determine to which access point one is connected. It's usually not a problem for most roaming users, but be prepared to deal with it.
I would suggest you install 12 access points and one spare. Leave out the 13th.