You can get the sheets of thin cork at most craft stores, even sometimes at dollar stores.

The humidity, well, I've never had any luck with keeping the house at a good range when winter gets cold.  Mostly the instruments spend more time in their cases then, with an in case humidifier.  Nothing fancy here, I use a pill bottle with holes drilled in the top filled with those silicon "water marbles".  I take the bottle out of the case and fill it up with water when I start morning practice, then pour off the excess and put it in the case when I put the violin away.  According to the hygrometer I keep in my case, that usually keeps it close to 50% humidity inside the case.  

It takes a while for low humidity to strip much moisture from the wood of the violin.  The finish has the outside covered, and the amount of wood that is actually bare on the inside is only open to the outside air at the f-holes.  So an hour or two of practice or an hour of playing here and there through the day won't do a lot to the wood, so long as it is kept at somewhere near the right humidity in the case.

The tuning pegs, well, it depends on how you like to do things.  Some folks like to keep them very tight so they don't move easy at all, because they mostly use the fine tuners.  I peg tune every day, since I didn't put fine tuners on my acoustic violin.  Tuning them every day, I think that I kind of automatically adjust for the big changes in humidity, since I keep the pegs pushed in just a tiny bit tighter than the point where they'll slip.  I haven't run into any problems with the pegs.  Maybe it is a matter of how well the pegs are fitted in the first place, but since my violin was a 170$ beginner package, I rather doubt it is any paragon of peg virtue or anything. LOL   

But they don't usually slip, nor do they "creak" or stick.  I don't use any chalk or soap or drops or compound on them.  It is a fairly new instrument though, only had it about 2 yrs.  Maybe antique instruments have more troubles? I wouldn't know.

Another thing that might make a difference is that I tune every instrument in the house that can be tuned every day.  Even if it is still in tune from yesterday, I detune each string a little and then bring it back up to pitch.  I was taught to do that with guitars, to "equalize the tension" from any changes due to humidity or temperature changes.  I don't know how true all that is, but I've never had the problems I've seen other people have with keeping guitars in tune, so I've assumed it works.  Never heard if one should also do it with violins, but I do it from habit.

Once it is tuned up in the morning, it is usually good for the day, unless I change the tuning to GDgd for some songs or something like that.