Don't know what to do

If you have replaced the battery in the last year it should still be under warranty. Have it charged and tested by the people that sold it to you and replaced if need be. The alternator tops up energy taken out of the battery, if your battery is bad the alternator is constantly trying to replenish energy to the battery. Maintaining correct battery voltage is important for things to work properly but I thing the other two problem are unrelated to the battery problem you have. Make sure you have no parasitic draw on the battery ( a load left on even after the ignition key is turned off).

had the same problem on my 2002 tahoe I changed the battery and no more problems