6 Days On Southern Island

sorry, not a direct answer to all your questions. Don't know where you've heard that Dunedin is best place for party, but it won't be at that time of year anyway. Most students have gone home for the summer and of those left, lots will head to central Otago (near Queenstown) - ie Wanaka.
If I was you I'd concentrate on spending all 6 days around the central area if you're wanting a party (and picturesque scenery too!)

1. What is the best way to see Queenstown and Dunedin in just six days? i.e. coach companies, timescales for travelling, etc.
2. What are the best hostels to stay at?
3. What beautiful sites can we encompass into our time?
4. Any good companies to do a skydive with in either of these places?
5. Best places to party
6. Anything else that is a must on the South Island?

1)I agree, I'm from Dunedin and find it HORRIBLY depressing in the summer as it is a ghost town. Do not come here if you only have 6 days!
3)Spend 6 days in and around Q-town. Check out Wanaka, and DEFINATELY save a sunny day driving to Milford Sound.
Hopefully you are flyiong direct to QT, otherwise its 5 hour bus from Dunedin or 6 hour from Christchurch. Milford Sound is a 4 hour one way drive, but has the best scenery in the country, well worth getting up early, drive there stopping at all the little scenic lookouts etc, mope about at Milford and get a boat ride or something, then drive back.
2)There are a bunch of hostels in QT, just get anything that is in the town though (and book early)....taxis are expensive there.
4)Do a skydive with nzone....I DID!
5)Everywhere!
6) With only 6 days, I'd just chill out there. If you can hire a car (quite cheap in NZ) get to Milford for a day and Wanaka for another. The only other must on the south island is a drive up the west coast to Abel Tasman National Park, but you'd need another week to make that happen.
ENJOY!