I agree with Gary, go with a plinth block at the base of the door 5/4, or maybe in your case thicker, so you can butt what looks like shoe moulding int the plinth block cleaning up the need for returns or excessive trim work that just looks busy.
Q #1 the best way to fit a corner is take a couple of 2or 3 foot scraps and figure out what angle cuts look best in that inside corner, framers rarely leave a sqaure corner and drywall mud makes it more difficult. Put your tape measure away and fit each piece with a pencil and scribe marks, sneak up on your cuts until you are good at knowing how things fit.
Q#2 If the space is to small between the wall and the door casing then run your mouldings into the door casing as far as they will go sometimes to get a tight fit you will have to shave the backs off of your moulding. Sometimes it is just not worth trying to make the piece, keep your fingers in tact, they are only useful attached to your hands.
Q#3 Cope cuts produce the best results on inside corners no matter what the angle
Q#4 Caulking works, if your trim and walls are painted then caulk with white, let dry and touch up paint along the edges. If you want to use stained wood trim then use clear drying acrylic caulk applied to the back of the trim (or goo it in after the trim is installed if you know how to handle your caulk) so when you nail it to the wall it pushes up on to the painted part of the wall and then you can use a finger to tool it out. After it dries touch up the paint with a cut brush, you should be able to get a nice sharp paint line where the trim meets the wall.
Q#5 Plinth block, the first part of my answer and Gary explains it as well.
you may also want to get rid of your shoe moulding as well, usually it has been installed to hide a flaw with a previous install, maybe with your flooring?
Good luck with your project.
-- I found the board stretcher... finally!