Can a solid capacitor be damaged by a heat gun?
What exactly do you mean by "heat gun"?  If this is not temperature controlled, like a hot air soldering station, then you probably damaged a lot of things, not just the capacitor.  Obviously this thing made air hot enough to melt solder, but you don't know how much hotter.  If it is not temperature controlled, then using it to solder was a really bad idea.

The fact that a nearby part got hot enough to discolor means you really overdid it.  Everything in the vicinity of the IC you abused should be considered suspect and replaced, including the IC.  Even if a part appears to work, it could be damaged in subtle ways and cause hard to diagnose problems.

You ruined a bunch of parts and possibly the board under them.  Replace them and move on.