As for the turning off after a few seconds, you battery may no longer be fully charging and may need to be replaced. As for the Max button being stuck, it seems to be a bit of a design flaw. Air escapes from around the button when the vacuum is on and thus it seems to push fine dust out of the small gap around it, making it stick. I’ve had a little bit of luck using silicon lubricant on the button and working it into the gap with compressed air. This make take a few cycles but it eventually freed up for me. If you get the button unstuck and the vac still shuts off after a short while it’s likely a depleted batter.
It sounds as though you have a bad battery. I have noticed that my iPod and iPhone do charge more quickly when plugged into the car. It seems as though you are able to give the battery a quick boost but the capacity is depleted. If the guy you bought it from is unwilling to repair it or you don't want to go through the hassel of dealing with him, you may want to tak eth opportunity to pickup a battery kit from here and try the repair yourself.