Chemical treatment requires about 4 hours to make water safe to drink - the filter takes seconds. The filter shouldn't be allowed to freeze, but that's what pockets and sleeping bags are for. I've used our Sawyer filter without incident on several sub-freezing backpacking trips down to 10 degrees F.

Boiled water I believe is safe to drink immediately after boiling.

Skip the Sawyer mini and get a Sawyer 3-way or Sawyer Squeeze which will keep you happier. The 3-way is nice because it has built-in fittings for quick-disconnect hoses. I put quick-disconnect adapters in the middle of our bladder lines and refill the bladders without having to take them out of the packs. Bladders aren't great in the winter though - the hoses froze up in our last trip and one of the plastic quick disconnects ended up cracking from the expanding ice inside. I hear you can blow air back up the hose after you take a drink to prevent this sort of thing, but I'll just use a titanium water bottle next time I'm out in the winter.