Can living organisms sprout in a dirty fish tank?

What you're seeing is planaria. They're pretty harmless but tend to show up when a tank is dirty. Just clean out the tank and vacuum the gravel to get rid of them.

If they are thin like thread (or even skinner than thread) then they are nematodes and, yes, they like dirty tanks and a low water current. They are thriving in all the stuff in your dirty tank. If they are a bit thicker than thread, and some may have a wee arrowi-ish head, then they are planaria and also love dirty tanks. These are rather hard to eradicate once they establish themselves because they can multiply both by breaking off pieces of themselves, each piece generates into a whole separate planaria, or by the eggs they lay. The more things they can feed on the faster they multiply. How are your poor fish?

when you have a dirty tank, its usually algae. you can buy algae eating fish (chinese algae fish-you can find it at petco), so you wont have to clean it up yourself. thats what i always do :).