Situation: runner at first, one out: Short pop up to first baseman. Ball takes one bounce. First baseman grabs it touches first base and tags the runner who didn't leave first base. Is it a double play or is runner safe at first?

Runner is safe at first. As soon as the base was touched the runner is not forced to leave the base. He should have tagged the runner before touching the base.

one other thing an umpire must judge if the 1B had done it correctly and touched the runner 1st then the bag is whether the 1st baseman intentionally let the ball drop to try to turn a double play..

8.3 J BATTER RUNNER IS OUT
When an infielder intentionally drops, or lets drop, a fair fly ball, including a line drive, that can be caught with ordinary effort with first base only, or first base and any other base(s) occupied, with less than two outs. A trapped ball shall not be considered as having been intentionally dropped. EFFECT: The ball is dead, the batter-runner is out and each runner shall return to the base occupied at the time of the pitch