When a 'palette' is used, does that mean that only those exact colors in the palette are used? Or is it possible that they can be mixed together or changed slightly?

There's a short and a long answer to that.  The short answer is that that's often what it means.  The long answer is that it kind of depends.  The Liberated Pixel Cup "palette" was a set of base colors that were intended to be mixed together and tweaked, rather than an exact set that people were meant to use exclusively.  The Dawnbringer palettes are meant to stand on their own; that is, they're designed in such a way that it shouldn't be necessary to use colors that aren't in them.

Is it a good idea to tint some of these sprites in a game for recolors of monsters and items if the sprites follow a specific palette?

It can be, but often times the palette will have a pretty big effect on how the art is made.  It's not necessarily a simple thing to convert pixel art made in one palette into another palette, particularly if one of the palettes is a lot more limited than the other one.  In short, it's certainly okay to try doing this, but I can't guarantee the result will look good.

Is it a good idea to mix some sprites from this tileset with another tileset (after resizing the sprites to 32x32), ie. http://opengameart.org/content/dungeon-crawl-32x32-tiles?

If you're trying to mix that with the 16x16 set you just linked, you might find that it's a bit difficult to do.  Just tinting the existing colors will probably look kind of strange and out of place (the large number of colors on the 32x32 set will clash with the small number of colors on the 16x16 set), so you'd almost have to go over the entire 32x32 set with flat colors and reshade it.  I'll post an example of what I'm talking about in a bit.

Here's what I'm talking about.  Note that due to there being fewer colors, I had to flatten out some areas of shading and do some dithering (which works okay on a strawberry, but other items may require flat shading).  It's not just a matter of replacing colors.

(Interesting note: I asked some pixel artists the same question a couple weeks ago about a different set of tiles and they told me the same thing I just told you.  This pretty much demonstrates that they're correct.)

This is good advice to follow carefully bart. I also lack creative talent so my solution is to ask a pixel artist when I have questions. This is a good place to ask questions since many pixel artists and developers like to visit this site.

As someone who is also not an artist, the axiom I go by is 'consistency is king'.   If all your assets use the same pallete, they'll look nice together and the overall image will look solid and professional.  Start mixing palletes and you'll get an image that looks busy, jarring or worse sloppy, even if the individual pieces are, by themselves, stronger.   Same can be said of shading, perspective, artistic style, etc.

Bummer is it makes it harder to mix and match art from this site, as Bart's strawberry demonstrates.

So, I just ran that tile set through Photoshop and converted it to DB16.  (I also tried GIMP, and the result wasn't nearly as good).  I find that the large amount of dithering clashes with the 16x16 tiles, but the result is actually a lot more usable than I expected.  Some tiles are more usable than others.

I find your strawberry ammusing, Bart.  The second image may not be very red and lost some of the nice dents around the seeds, but it's actually much better from a color theory standpoint and would look really good in the right setting.

I'm pretty surprised with how well that converted too.  It needs a little work but looks pretty good and would take a lot less effort to fix it than it would to start over.

What we all should learn from this is that a good artist can even make existing art work with a different pallet and that Dawnbreaker is awesome.