First you need to check the laws in your area. In MIssouri, it is not illegal to leave a child in the car while you run in to get a gallon of milk. The law steps in when there is injury to the child or caused by the child (say they put the car into gear and ram into someone).
And being left at home is up to the parent's discretion.
Other states are very strict about the ages a child can be left alone and for how long. SO, you need to check and figure out what your local laws are.
As far as his behavior is concerned, I would keep documentation of EVERYTHING. Also, figure out what the laws are in your area when it comes to recording conversation. In Missouri, only one person (you) is required to know that the conversation is being monitored. Get a digital recorder and record every conversation you have on the phone or in person. Download them and save them on your computer. When you feel that you have enough documentation and recordings, go to your attorney and file for a modification of support and custody. Require that his visitations are monitored, if he is to have any at all.
Best of luck!