To sell something to him, right click on it in your inventory, or drag it to his window. You automatically receive the AirRivals gold. If you accidentally sell something, the last thing you sold stays in a slot on the bottom of the vendors window that lets you buy it back for the same price you sold it at. Be careful what you right click on when vendor window is open.
To equip an item you bought or looted: Open up your bag and right click on the item. It will automatically switch the item out for you. You can also open up your inventory and drag and drop items.
You can share quests (give them to another player) in your party if and only if it is an open quest. For example - If I run into a city Ive never been to, and I have 5 available quests without doing anything, I can share any of those with my party. If I do those 5 quests and they unlock 2 new quests that follow the first ones, I cant share those within my party, unless the other people have completed the pre-requisite quests. This is how I understand it.