I have a standard 200 watt power inverter which I have used to power a sound system off of my Econoline e250 on multiple occasions with no problem. I tried to run a 4hp rigid shop vac off of it and the cables to the inverter got so hot they melted the plastic on the top of my battery a bit. Is there a chance I could have damaged my alternator?
To be direct, yes it could ... but most inverters I've seen are fuse protected and/or have a thermal cutout on them. If the inverter was drawing too much juice, it could possibly damage your alternator, but it's highly unlikely unless there was an unseen issue with the alternator in the first place. If it did damage your alternator, I'd highly suggest you get a different inverter as it will cause issues for you in the future. If it did cause alternator damage, the built in safety devices failed and will do so in the future. That it worked at that much wattage in the first place is beyond me.
Yes, it could, but you said the car would then not start or charge, and seems OK now you have installed a new battery. In my opinion it's more likely that you have damaged the battery. Batteries are designed to supply hundreds of amps but only for brief periods - you asked it to do that for 15 minutes.