Yep. Just buy one like this at any hardware store. Easy to replace. Black wire (hot) to copper screw, white wire (Neutral)to silver screw, and green wire (ground) to green screw. Make sure they wrap around the screw in the same direction that you tighten.
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So replacing the plug can be done, but doing so is sometimes against local electrical code (so can't use it in places like businesses). If it is for a home machine, then that is no real concern.

Yes, you can do this - even in a commercial environment as long as the color codes are matched to terminals and you do a good job (e.g. tight screws, no cut or loose hanging strands).

Also note that on some of the newer power cords, the colors can be the new IEC colors:
Black wire (older cords) = brown wire (newer cords)
White wire (older cords) = blue wire (newer cords)
Green wire (older cords) = green wire with yellow stripe (newer cords).

Yes, though I generally prefer to replace the whole cord with a fresh 14' cord, since original cords typically have minor age or wear issues present.

I have a couple of these plugs on-hand: https://www.amazon.com/gp/product/B0015P708Y

Also, make sure to use the multimeter to test continuity of the grounding prong all the way into the game.