Go to www.diptubesettlement.com and see if the model number of your hot water heater is listed. A telltale sign of a broken diptube is little white plastic pieces found in your faucet aereators or in the bottom of the tank. Your wh is past its guaranteed life but if it isn't leaking you are ok.
Are you saying you don't have "any" single handle faucets or just not one in the bathtub. As I said before, it could be the shower, tub, lavatory, or kitchen faucet.
A water softner will help with the hardness.
I'd want to think further and we'll see if any others have an idea. After you've replaced the element, then post back in a couple days to let us know if you are getting the hot water you are supposed to. By the way, you said you are replacing one element, what about the other element and the thermostats that run the elements?
Remember to throw the breaker and don't energize any elements until they tank is refilled with water or the element will be fried.