Water is not what we are referring to when we say cycled.
Cycled means having bacteria that eat ammonia and nitrite growing on your filter media (the spongy stuff). Otherwise, the waste from fish adds up very quickly to toxic levels.
Old water is never a good thing. :)
And even with cycled tanks, we do water changes to remove nitrate (the by product of bacteria) and other bad stuff.
I'm very glad you have a heater, I must have missed that line. Also, we aren't all American, all are welcome. :)
There are anti-intestinal parasite foods out there to try. Also, sometimes their bellies bulge from constipation...either way, a pea shelled and cut to bits is good for them every once and a while...cleans out the system. If the water stays pristine clean, then the ulcer should go away.
If you don't care about the snails, a bit of aquarium salt in the water may help the betta heal, but will hurt the snails.
10 gallon with a veiltail betta, 7 neons, 1 ADF, and one neat moss ball.
20 gallon with 6 female black phantoms, and 1 schwartzi cory