Can a 7 speed Derailleur work on a 8 speed cassette I have a extra 8 speed shifters? and a wheel that has a 8 speed cassette.
8 speed shifter and cassette, 7 speed derailleur? You'll be fine. There's no "probably" about it.
6-, 7- and 8-speed gear is all cross-compatible in terms of chain width.
The only time you'd hit a problem is if the shifter and the cassette weren't made for the same number of gears, for example you could imagine a scenario (e.g. 7-speed cassette, 8-speed shifter) where you might click past the edge of the cassette. That would negatively impact your day.
Probably. Depends somewhat on whether the shifter is compatible with the derailleur (Shimano 8-speed shifters generally will work okay with 7-speed derailleurs, but you can't usually do a Gripshift with a Shimano derailleur, to name a couple of examples.)
You'll have to adjust the limit screws and barrel adjuster, but you would have to do that any time you install a different derailleur.