It sounds like you have a sidekick sport. The tranny you picked up was from a regular sidekick/tracker.
Simply put the bellhousing off your tranny on the new tranny and it will bolt up and work fine.
You will also have to use your tcase on the new tranny, if the new tranny came with a tcase, since the new tcase will not have the correct spline output shaft to fit your rear driveshaft.
For the second problem, it sounds like something is binding in the drivetrain....could be the tranny, could be the tcase, could be the axle, could be the brakes. You are just gonna have to look everything over, spin the driveshaft by hand, check for looseness, spin the tires....check to see if you see or feel anything binding, or if anything is excessively loose, see anything that doesn't look right, etc. Jack the rear end off the ground and block it up, so you can rotate everything and look it all over.