Actually, if we dig deeper, I think it should be: No, sort of, it
depends and Yes.

1 - Can I buy a 3 prong power cord and just attach the ground to the
case of the power tool?

No. Just attaching the ground wouldn't do much for you. You would also
need to attach the Hot and Neutral wires to their proper places. ;-)

CY: She could ground the three prong plug, and leave the existing cord
in place with the two prong plug.

2 - Is this what the 3rd prong does?

Sort of. The 3rd prong actually connects the grounding wire of the
power cord to the receptacle's groung lug. Correctly installed (the
cord, the plug and the receptacle), this should connect the
equipment's case to ground, which is the ultimate goal.

CY: Works for me.

3 - Would this make is safer?

We would need a little more information before we can properly answer
"Would this make is safer?" It might...it might not. If the tool is
already double insulated, would a ground wire really make it safer?

CY: I'm with you. Might, but likely not a lot of advantage.

4 - Does a GFCI also work on a power tool with just 2 prongs?

Yes

CY: Agreed.