It's a circa 1979 Miyata Gran Touring, which was the predecessor to the highly regarding Miyata 1000. Almost everything has been replaced. Only the crankset and possibly the headset, handlebars and strem appear original. Even then, the middle chainring is a replacement.

It's impossible to tell is the nut is stripped or not, You will just have to try it with a new cable.

The Deore LX had a limit of a 32T maximum cog. Often, you can exceed this by 2T, so a 34T MegaRange may work, depending on things like the length of the hanger on the SunTour GS dropouts. Again, the only way to know for sure is to try it.

The frame was originally a five speed. The rear triangle has been cold set if the spacing is 126mm. You should have no problems running a 7 speed freewheel as both 6 and 7 speed using 126mm spacing. The overall width of both freewheel bodies is very close, with the difference being slightly narrower cogs and spacing on the 7 speed.