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Runner on first.
Runner steals
Batter strikeout, but catcher dropped the ball.
Batter runs to first and runner continues from second to third om an error from the catcher. (bad throw to second)

Umps called the batter out saying the first base was occupied.

Two questions.

Is this correct? Should the batter have had a strike out since the runner did not occupy the bag? He was over halfway to second.

Does the runner then get two steals? 1st to 2nd and 2nd to 3rd. I guess the 2nd to 3rd would be from the error. 1st to 2nd he was there in plenty of time anyway.

Yes, the call is correct, assuming less than 2 outs at time (sounds like this was the case since the runner advanced to 3rd) and I actually saw the same play this weekend watching NCAA baseball. Even though the runner was stealing, at the time of the pitcher's delivery, the runner was still technically occupying 1st base. The base occupied by a runner is defined as the base he is legally in possession of when the pitcher begins his delivery. Stealing, or not, the runner still has the right to return to 1st base any time before he obtains 2nd base. However, if there were 2 outs (sounds like there weren't), then the runner at first IS forced to advance when the 3rd strike is dropped.

You are correct on the steal, the runner is awarded 2nd base as a steal, since he began his steal before the PB or WP. But, his advance to 3rd was probably only by reason of the PB/WP, therefore the advance is credited as such.

TurboAg nailed it that the runner's base is from time of pitch. I wanted to add something regarding the steal from first to second. Camerabry mentioned the runner was there in plenty of time. I just wanted to follow up and say that it wouldn't matter. Even if the throw is ahead of the runner but is off target, the runner is credited with a steal. The catcher doesn't receive an error on a bad throw if the runner is stealing. He would only get an error on the advance to the next base (third in this case).