What a problem could cause antifreeze to get into the exhaust system

What is your question?

First I had a battery problem I left the light on and the battery was dead recharge the battery and drove approximately 825 miles and it wouldn't start again so got a jump took it home and had the battery shop put in a different battery however it was the wrong battery for the car the next thing I know is it tells me it's overheating but it wasn't hot so I took it to the shop and they said it was a blown head gasket but they're just do peer to be anything wrong with it now the only thing that's going on is the check coolant light keeps coming on in the check tire pressure light

How long has your 2005 Cadillac DeVille had this problem?

Happened once and then I was told it was a head gasket and I need to put in a new motor