kylehutson

Assuming you're talking NFHS (high school) rules...
Yes, but once he does, he can no longer be a courtesy runner. A courtesy runner cannot have participated in the game in any other capacity.

A foul tip (directly to the catcher's glove and CAUGHT by the catcher - just to make sure we're on exactly the same page) is a live ball and treated exactly the same as a swing-and-a-miss by the batter. The ball is still live, and a runner stealing can continue just as as before (i.e., option "C")

Caveat: I *think* softball is the same as baseball, but I'm no expert on that. (You didn't say, specifically, but the use of "she" makes that a likelihood.)
Batter-runner gets first base. All other runners go back to the base they occupied at the time of the pitch, unless forced by the runner getting first base.
So in your case, the batter-runner would go to first. R1 (the runner that started on first base) gets second.
But if you had a runner on third, also, that runner would return to third.

When the ball is caught in flight by a defensive player, the player has to "tag up" (i.e., touch the base they were on at the time of the pitch).[1] You typically see this when a ball is hit to an outfielder, and the runner leaves the base believing it to be uncatchable, then having to return to the base. The touch of the base is an appeal that he left the base early. The same holds true in your scenario, it just happens more quickly.
[1] OBR Rule reference 7.08(d): "Any runner is out when he fails to retouch his base after a fair or foul ball is legally caught before he, or his base, is tagged by a fielder. [...] This is an appeal play;" Other rulesets have similar language.

I don't know a lot about softball, but that one difference I do know. If you don't pull back the bat on a bunt attempt, it's a strike.
In this case, dead ball (because it hit the batter), strike (because the bat didn't get pulled back), and out (on strike 3).

To expound... Any time a batter swings and it hits him before hitting the bat, it's a dead ball strike. It's not uncaught, because the ball is dead. Therefore the batter is out.

Feet position make no difference. It's whether or not the ball is foul at the moment the ball is touched. 99.99%+ of the time, that will be what it would have done had it not been touched, but one could imagine a scenario where the ball is curving relative to the base line, where that wouldn't be the case (but very unlikely on a pop fly).