If a batsman drops his bat while running, and crosses the crease without it - does his team still get a run?
Similarly, if he makes it passed the crease, if the fielding team hits the wickets, is he out since he does not have all his equipment?
If a batsman drops his bat while running, and crosses the crease without it - does his team still get a run?
Yes. Law 18 of Cricket makes no mention that the bat has to accompany the player for a run to be completed, simply that the batsman has to be grounded.
Similarly, if he MAKES it passed the crease, if the fielding team hits the wickets, is he out since he does not have all his equipment?
No. Law 30 states the following:
A batsman shall be considered to be out of his/her ground unless some part of his/her person or bat is grounded behind the popping crease at that end.
It seems that the only downside to dropping your bat while running is that you simply have slightly less reach to ground yourself with. Beyond that, the bat is mostly not necessary for running. I was expecting to find a rule regarding equipment being left "in play" on the field, but it seems that these laws only refer to the fielding side's equipment, and not the batsmen.