Is there a connected plumbing problem if both toilets in the house aren't working properly?

We have 2 bathrooms-- one upstairs, one downstairs. Recently, the toilet upstairs started running for a long time after flushing. Then, just yesterday, the toilet downstairs stopped flushing at all. The tank in the downstairs toilet is basically empty, and part of the machinery in the toilet tank is leaking water. I am wondering if these two problems are related because, for one, they happened at the same time, and for two, we have a problem with corrosion to our pipes etc. The showers have water pressure problems (lack of) because of buildup. When we replaced our water heater, the water pressure in the shower downstairs practically slowed to a standstill (still haven't had that problem fixed because they have to cut a hole in the neighboring bedroom wall to get to the plumbing to fix it). Anyway... anyone have any ideas? Thanks.