This is sort of answering my own question, but the only resource I could really find as far as how long blood stays fresh was information on blood banks, here's a wikipedia link for those curious.
http://en.wikipedia.org/wiki/Blood_bankIt just seems that regular folks aren't interested in the stuff after it has left the body in an unsterilized environment
It says that blood has just a little over a month's shelf life before it is considered unfit for transfusion, with the blood probably needing sterile packaging and a refrigerated environment for that to be true. I still don't know the rate at which blood spoils or hardens specifically at room temperature, and I don't have the slightest idea where I'd go about finding that out, but it seems that if one really wanted to go about making a blood fountain without having to drain and refill it periodically, they'd have to do it in a very cold, sterile room, with the fountain itself having probably been purged with fire and antibacterials.
I'm sure that very few people are keeping their fountains around as an emergency blood bank, and if its not allowed to sit, the blood probably has at least a day of being pumped around at high pressure before it becomes nothing more than a stinking black mass, but that's still just a guess.
I also have no idea how air exposure would specifically affect just blood, as I'm not entirely sure clotting is facilitated only by the blood stream...
If anyone else has any thoughts or knows more on the subject, I'd be glad to hear about it.