1. You need to go to the options menu (possibly camera menu - I forget if there is one), and turn off director mode. There are lots of options where you can choose whether or not the game pauses or centres on enemies or characters when something happens. I recommend turning off a few of the default settings as they are annoying as hell after a time.
2. You must remember that every action in the game takes up a number of action points before they can be performed in real time mode. When you change weapons and items and go back to the game screen, you will see the character's timer displaying a few numbers. This is how long it will take to perform your current action. You need to unpause the game and wait for the counter to tick down to zero before issuing a new command or he won't swap weapons. This takes a little while to get used to but is a crucial part of the gameplay.
3. You need to drop weapons on the floor for others to pick up. I'm not sure if you can pass items to one another by throwing them or not. Been a while since I played.
4. Don't bother carrying aliens. Sometimes you can't carry any aliens because they are too big. Instead kill all but 1 alien in the mission and you will win the mission anyway. Just remember when an alien is KO'd to cancel firing orders so it isn't killed, as killing all of them results in a defeat.
[ Edit - If you DO want to carry a live alien once it is knocked out, select your soldier, then press M for Move, and click where the alien is lying. Your trooper will move over the alien instead of shooting it. Now go to the inventory screen and you can pick the alien up. ]
5. If the squad is currently travelling you can't move a soldier across from the pool to the main squad. If he isn't in training but can't move across and the squad is still at the base, make sure he isn't wounded. Wounded soldiers are out of action until healed.
6. Not sure about this, you may have to unequip the current weapon before equipping the new one. Certain weapons require heavy armour before you can carry them too. I think the game is a little pernickety at times about how you do things. Sometimes if you don't place the item EXACTLY in the space it will snap back as you say it.
Hope that helps.

Yes, AM has some bugs. First, I recommend downloading the ComboMod Beta 3. No, your current savegame won't be compatible with it. Sorry. It changes the game by fixing some balance issues. You'll have to read ComboMod's readme, to see all the stuff it changed.
I only have that happen once per alien type, once per mission. Doesn't the camera return to the default view when done?
In my experience, attacking / reloading orders override movement, and movement orders override rummaging in the backpack. You'll have to remember which folks are digging in their packs. Oh, there's a suitcase icon when they are rummaging, and an up arrow when they are reloading.
I believe you can press the T button to throw what is in your hand at another character. My preferred method is one dropping and moving (which is a little tricky), and the other moving and then picking up. I'll try throwing it next time.
Like Collie said, just kill them all, its easier. However, there is one mission where you have to carry a box to an extraction point. Make darn sure that box is in your hands when you cross into the extraction point (and make sure that character is last). Then you will get the mission completed. Otherwise, you may hit the 'abort mission' button, have it be successful, and never get the part of the tech tree you need. Arg! Be sure to fly directly back to base, too! If you have already inadvertently completed this mission, let me know. There's a way to export your characters to a new game. Its not cheating when you have to start the game over to get the research to need to finish it!
If there is a helicopter icon, your people are out on a mission, and the new folks can't xfer until they land again. If you've just conquered a new base, hit the play button (on the geoscape, the slowest one) a few times. This will give you the three messages:
1. You've conquered a new base (influence, blah, blah)
2. You've got new exciting crap (err, equipment)
3. Your helicopter has landed (yay, you can give your folks the new crap, cycle them out for hospital/training/what-have-you)
See #5. By the way, when your folks are not out, you can double click to quickly add ammo to the current dude. You can double click the ammo/items if you add too much to send it back.