One-way rentals are not a problems.

I'd recommend you pick up the car on the West Coast, drive to Queenstown and then drive to Christchurch. The drive down the West Coast is amazing, as is the drive from Queenstown to Christchurch going thru Central Otago and Mackenzie Country... the only problem I foresee is not having enough time to do all you'd want to do along the road

I wouldn't worry about road conditions. While snow is certainly a possibility, Highways Six and Eight are key trunk routes and thus expected to be kept open except in extreme conditions.

(BTW, my trip report linked below will include sections applicable to those areas when I complete it).