There are several things at work here.
1st, when you kill the battery the ECU is losing its memory. This will make things like idle and such a little wonky until it relearns itself.
2nd, killing the battery several times could have in fact weakened it. It being in a weakened state could cause all sorts of electrical issues.
3rd, the fact that running the radio for only 20 minutes kills the battery tells me it is weak. You need to have the battery tested at some place like Advance, Auto Zone, etc (whatever parts store is in your area) They will be able to tell you if the CCA's are within range and if there is proper voltage.