"...etc." is short for "et cetera". The "et" means "and", so it makes no sense to write "and etc". Just use "etc". "...and stuff" is a very casual expression. You'd never use it in serious essays or other writing, nor in a proper speech. It's OK in casual conversation, but it also sounds like you don't care much about what the "stuff" is. The useful expressions are "...and so on.", and its longer sibling "...and so on and so forth." I recommend using these, because "etc" is also an abbreviation; it's not generally acceptable in proper writing, and it sounds a little odd when you speak it (as if you were reading from notes).

"and etc." is not quite correct. You would normally say just "etc.", without the "and". "and stuff" might be used from time to time, but "and so on and so on" would be used more often. "For school, you need books, pencils, etc., etc." "For school, you need books, pencils, and so on and so on."