First: if you use one, what do you call the thin object that goes under a cup to protect the surface beneath it (usually a table)?
I grew up with “Deckel,” literally a little cover and usually denoting a lid, which if you think about it is counterintuitive: why put the lid of a cup under the cup? Semantically it’s an all-purpose cover, insofar as a Decke (two syllables) is a coverlet for one’s bed—and there we have the diminutive endings reversed, eh? (Coverlet.) In usage, the Deckel I’m used to doesn’t resemble a lid; it’s a piece of pressed cardboard, usually square and usually advertising some commercially produced beer. Then again, my father always used one to cover his individual container (cup/glass/mug) of beer, juice, wine, or water when walking away from the table for more than a moment. That makes it a lid against flies, at least.
Local usage dictates “coaster,” otherwise, which to me is as unexpected as a cup-coverlet because coasting implies motion and the last thing one wants is for the cup or the thing over/under it to creep away on its own adventurelet.
Anyway, the little cross-stitch kit I mentioned the other day is a set of four “flower cup coasters,” which here means that the acrylic coaster halves enclose cross-stitched depictions of little mugs (not cups, ahem) with flowers growing out of them. Since they’re so quick, I think I can stitch all four in time to give my mother and darkforge’s mother a pair each for xmas. Here is the first one, not yet endeckled:

(Cup versus mug: latter is more specific—larger, often thicker-walled, with a handle.)
Coasters. (But they were Deckel when I was in Vienna. I collected a bunch of them.)
Ah! This is grist for my supposition (in a comment over on DW) that “Deckel” in my father’s usage is either dialectal or outdated. :) More likely dialectal, I think, though since he and his brother were born mid-1930s, I couldn’t rule out the temporal possibility without more data. Thanks for this!
First: if you use one, what do you call the thin object that goes under a cup to protect the surface beneath it (usually a table)?
I don’t, but my family always called them coasters. I have no idea where the term comes from.
(The OED: “Formerly, a low round tray or stand for a decanter (usually of silver); now used generally for any small, usu. round, tray or mat on which a bottle or glass, etc., may stand. (See also quot. 1890.) So called from ‘coasting’ or making the circuit of the table after dinner.” Okay, that’s actually more interesting than I would have thought.)
*nods* The OED sense seems to describe something a bit larger than what the US usage tends to denote, however, no?
I will say, sadly, that while we have a lovely collection of coasters, we had to put them all away once the kids got to the pulling-up-stage, as they became throwing toys par excellence.
*nods* This is one advantage to my not being a Real Grownup: less stuff to break. Also, doilies and cardboard squares are not very satisfying missiles.
I use beer mats instead of coasters for my mugs – both at home and in work.
Beer mat is essentially what I grew up calling a Deckel. I still have a few cadged from long ago.