Hopefully that's all of them:
1) Depressed in Widow nightmare.
DOWN (4) - Depressed in WiDOW Nightmare [[found by Duncan]]
2) Star lit up rebel city.
CELEBRITY (9) - anagram of 'rebel city'
3) Former wine "contains exhaust"
RETIRED (7) - TIRE means to exhaust, surrounded by RED as in red wine [[found by Gareth McCaughan]]
4) Wicked "back to walk the earth!"
LIVE (4) - when reversed becomes EVIL and only living things walk the earth (NOTE: This clue works with either LIVE or EVIL as the answer.)
5) Moral within "Marathon" establishment?
HONEST (6) - within "MaratHON ESTablishment
6) American state upsets old fair
FLORIDA (7) - anagram of 'Old Fair'
7) Tree droppings, "go away!"
LEAVES (6) - fall from trees and LEAVE is to go away
(Suggestion: This clue ought to be tweaked so that the two parts are either both plural or singular, possibly something like: Trees do this: go away! with an answer of LEAVE (5)
8) Small discount mound (after looking at the clue)
HEAP (4) - Discount means CHEAP, 'small' suggests shortening the word to give HEAP which is a mound.
[[Note: Just as I'm about to press save edits, I've noticed that Jonathan has this answer as well]]