How much does it cost each time you skip a timer?
What Dave said is pretty much right. 20 gems when it first starts running, reducing by 1 every 40ish minutes. (It’s not exactly 40 minutes though. I know, for example, that it changes from 2 to 1 with exactly 54 minutes left on the timer.)
How many times can you skip chests per day?
Is the limit in question #2 per chest or total chests skipped?
You can skip up to three in a 24 hour period, starting from when you skip the first one. This is regardless of which type of chest timer you skip. (This, coupled with the fact that Diamond raid chests give really good loot, is why I only skip raid chest timers; the slow rate of raid flags provides a good timing/spacing pace.)
Does it matter which chest you skip?
No, I don’t believe so. I don’t think there’s a way to prove this though, without TONS more data than one person can collect.
How do you fill so many chests? Does it involve buying flasks? Which of the two chests is cheapest to fill, if so?
I rarely use flasks for filling normal chests. Maybe if I’m bored. I DO, however, use World Energy flasks when elemental chests do appear, both because I’m impatient to get the loot, and I want the timer to start running again ASAP.
What’s your fastest, average, and slowest wait time between elementals?
Stated in my comment above in terms of the # of chests, but in terms of days (since I began tracking), the shortest time was five days, and the longest was 13.