I am trying to put in an exterior outlet. I have a 20 amp breaker. The outlet that I am trying to pull power from has 12/2 wiring with a white sleeve. I am using 12/2 UFB wiring from the interior outlet. The outlet that I am trying to use is full. Can I take the white, black and ground off the bottom terminals and put a white, black and ground short wire on the terminals and then tie all three white wires together and then the three black wires together and then the three ground wires together and run the 12/2 UFB to the exterior outlet and wire in up. Also the exterior outlet has a 15 amp receptacle. Can I use that or will I need to buy a 20 amp since I have a 20 amp breaker? Am I wiring everything correctly or not? Do I need to use the same wire as the interior has or can I use the 12/2 UFB? Please help!

Yes, you can tie all the wires together with a wire nut and add a pig tail to connect to the outlet. You can used the UFB cable or NMB (Romex) assuming that the exterior outlet is mounted in the exterior side of the wall and the cable is not going underground anywhere. NMB (Romex) is probably easier to pull and cheaper. Your only restriction will be the size of the box where your are tying into. It must be large enough for the additional 2 wires. Using a crimp sleeve to connect the ground wires saves a little room over using a wire nut.

Assuming that "white sleeve" means white sheathing.

Exterior outlet must be GFI or GFI protected.

In addition to Harold's good info, code allows either a 15 or 20 Amp outlet. NMC can be used through an interior wall to service an exterior outlet which must be GF protected and covered with a weatherproof cover. If space is an issue, put the GF on a previous outlet and use the load side for the new outlet and put the label "GFCI protected" on the outside outlet.