Unfortunately your question requires a "that depends" answer as it depends on the battery charging system. I see in your signature that you have a 2002 Gulfstream Seahawk 29FKS. Being that old it's highly possible that the converter/charger system is only a one stage. If that is the case then you can not leave it plugged in without boiling the batteries dry. If you can look at the converter/charger for a sticker or label and tell us the make and model we can help figure out if it's a 3 stage charger.
As far as the tires it makes no difference if you store it on dirt, gravel, wood, asphalt or concrete. But you might want to put the landing gear legs on some wood to prevent gouging or rust stains in the concrete.