1. Die falls on the floor (but still visible, doesn't roll under the refrigerator, etc.). Stick with the roll or reroll on the table?
Reroll.
2. Cat swats the die but it still stops rolling where you can see the value.
If the die hadn't stopped rolling before the cat swatted it, keep the roll. If the die had already stopped, reroll. Then gently pick the cat up off the table and put it in someone's lap; that person is then tasked with keeping the cat occupied and thus off the table.
3. Golden Retrieve with Pica quickly eats (as in swallows) the die.
n/a as I don't have a dog. Bad idea, though.
4. Dice trays. Require, ban, don't care either way.
Don't like as they take up too much space.
5. Digital dice rolling for in-person games.
No, with the exception that if a really bizarre die has to be rolled to suit a chart or table e.g. a d-537 or a d-93 then digital rollers can come in really useful!
I want them clear and readable enough that they don't have to be picked up and squinted at. I've been having this argument with different players, who (for reasons ranging from aesthetics to nefarious) want to use the fancy unreadable ones, for almost my entire DMing career.
7. Large, heavy, loud dice (metal, rock, etc.)
Prefer not.
8. Precision dice that don't roll much.
Make them roll, or don't use them.
To me these are just the same as dice trays, taking up space on the board for no good reason. Lose them.
10. DM roll in the open or behind the screen?
Behind the screen, with rare life-or-death exceptions.
11. Roll against the DM screen like back wall at a casino craps game
Given that for space/layout reasons I DM from a different table than the players are using, that would lead to a stupid amount of dice on the floor. In other words, hard no.
12. How to punish bad dice?
I just threaten my dice with this, but I've known someone who actually did it: take the offending dice outside to a concrete pad and - with all the other dice lined up watching - melt them to slag with a blowtorch.
13. Using another player's dice.
Some people are way fussier about this than I am (see evidence, this thread). For me, go ahead and use 'em as long as I get them back right away. No table rule.
The only other dice "thing" around here is if you drop a die or roll it off the table or whatever, pick it up NOW even if it means interrupting the game while we all look for it. Particularly if it's a d4.