Grip shift won't move
You want to try moving the derailleur by pushing it with you hand, which will isolate whether the mechanism is stuck, or if the grip part is stuck.  If you have exposed inner wires, then try pulling them with your hand to see if the derailleur mech moves.

Could be corroded wires in outer sheaths, damaged grip control, or corroded/frozen derailleur.

Note that twist shifts/grip shifts were pretty low on the quality scale, and tended to indicate a budget bike.  You might be on a hiding to nothing spending money on this bike.



Chain skip sounds like worn sprockets on the rear wheel, an elongated chain, and possibly impacted dirt between the sprockets stopping the chain sitting on the teeth properly.

Consider posting clear and well lit closeup photos of the parts, which may help others answer more accurately.