I currently ride a 3x7 Tourney bike, which I’ve upgraded moderately and it is riding nice. I’d like to know if I can somehow replace my 3-crankset for a simple one, and also get rid off the front shifter and its cable.
Yes - note that some bikes will still need the front derailleur to help prevent the chain from falling off, which defeats the lightness goal, and only makes the bike slightly simpler.
You'd use the upper and lower limit screws to force the FD cage to an appropriate place, and then can remove the cable and shifter.
Yes this is possible.
Couple of options for single ring cranks:
If your current cranks have removable chainrings you can take off the outer and inner rings. You'll need short chainring bolts as the middle and outer rings typically share bolts, but these are available and inexpensive. You want to use the middle position for the single chainring as it aligns with the rear cassette the best. You can put the large ring in the middle position if you want.
A tourney equipped bike likely has a square taper axle bottom bracket. There are plenty of inexpensive single ring cranks available for this. Look for cranks for single speed bikes, or 'track' cranks.