2007 avalon electrical issue? (fob battery, car battery, car alarm)

vehicle was gift in april to my wife from her out of state mother. It was right after her job shut down as non essential. So, lacking the money for insurance and the 420$ for 1st time registration from out of state, it is not being driven. Ive driven it around the block a few times, but unless it's running at least 20 to 30 minutes, it does more harm then good. Car battery was dead the last few times, so now every few weeks im charging it up. yesterday the fob would not open the door. No beep. nothing. So, fob or car battery dead. I opened door with mechanical key. Fob logo held against the start button does nothing, so must be dead car battery. I open hood and hooked up and turned on charger. Immediately, the horn or alarm sounds. It never did that before. I remove the charger fast. I recover from the surprise noise and try it again. Again horn or alarm sounds. I disconnect battery from car and charge it that way. After a few hours, I remove the charger and reconnect battery to car and the alarm sounds. Now i know it was the alarm, cuz by the time I get cable off it had cycled two times.

What's going on here?

Wife says car don't have alarm. I show her the alarm button on fob. She says oh, we(her family) never used that.

ideas?

pa