Do I need a regular Tee or a Slip Tee to add new line (pex) to existing line (copper)?
Using a slip tee or reg tee depends on how much movement you have once you cut the pipe.  In either case you will need to make 2 cuts to where you want to add your tee.  The second cut will be made to remove enough pipe so that once your tee is set, you do not see any pipe when you look into the center leg of the tee.  I'm not a big fan of sharkbite, if it's not in a garage or somewhere that it can be inspected periodically then I would consider soldering copper.  The sharkbite slip tee will just be slipped all the way onto one end of the pipe and then there is a little tool you will need to back it the other way onto the mating pipe.  If you have an old piece of pipe for trial and error that will help you determine how much gap you will need to cut out.