It is possible the derailleur cage is bent or the new hanger isn't aligned. Maybe the new hanger was straight for production but not for your frame and someone slapped it on and didn't know or have the proper tool or didn't know how to use that tool and it is not properly aligned. It is also possible that the derailleur cage is bent which can happen when you crash or break your hanger. Sometimes yes it will save your derailleur but not always.

A derailleur should work for all the gears assuming the shifter is correct for the number of gears and the derailleur can handle the number of gears as well as the range. It is possible the chain length is wrong but that is unlikely to hit your wheel because of that.

This would be best in the Bicycle Mechanics sub-forum. However in your particular situation you may want to visit your local bike shop and have them take a look. I don't know if the seller was being totally honest or if they even knew what other problems that could have arisen from their crash. I would want a professional to look it over and see what is going on. Hopefully it is just the derailleur issue and easily solved but you never know.