Running toilet is likely the flapper. That's an easy fix, trust me. Watch some you tube videos, take the old one off, go into the hardware store and get a new one. The fill valve can also cause issues inside the toilet, that's a little more complicated but is totally doable as well. (I started w/toilets where you are, this is manageable.)
Mold on the wall:
What's the ceiling UNDER the toilet looking like? If you've got a leak, at some point you'll see water damage there. Water flows downhill.
In terms of identifying if the subfloor is damaged, the only way is to actually look at it. Some discoloration/staining is ok, rotting is not. If it's damp (aside from whatever water you splashed out of the toilet), soft, you can poke a screwdriver into it pretty easily, etc that's not good. How not good depends on how bad it is. You may be able to just let it dry out and fix the cause. You may need to replace stuff. And anywhere in between.
However, mold on the wall isn't necessarily coming from the base of the toilet. Simple condensation can cause mold. Do the sides of the tank get condensation? That may be the problem. You may have a leak in the water supply line (the flexible hose that attaches the wall pipe to the toilet). Those don't last forever - it's if 10+ years old, it probably needs to be replaced. Both of those you can handle. There's also the water pipe inside the wall (I'd call a plumber if that's the problem.) Where's the mold in relation to everything else? That'll help you find the cause.
Really, don't panic, and start trying to trace the source of the water.