Usually I don't drive longer than 15 minutes because everything is so close to me. Will I drain my battery and clog up my exhaust system if I never go for a long highway drive?
No, but your oil might collect some water. Try to do some highway driving to avoid that.
Is it bad to only drive your car short distances?
Yeah
Will I drain my battery
No. The battery charges in a few minutes.
clog up my exhaust system
what part of the exhaust?
The issue with short trips is that the engine doesn't get hot enough . Your engine likes best to run at operating temperature. This warms up the oil to its optimal viscosity, burns off condensate, deposits and stuff. The engine also runs in open loop until it reaches that temperature. That means the fuel mixture isn't finely tuned, and it's likely a bit rich. This could accelerate wear on the catalytic converter, and it also affects your economy.
