Which arena do free and crown chests use when determining the cards you will get

As a free-to-play player, there are two main things you can do with your gems to conserve it; make it last longer:

• Speed up chests that are nearly completed, not those that have just started.
  Alternatively, don't use them at all. You should be trying to save it for when cards begin to climb in costs (1,000 Gold at Level 8 to 9) and then [see below]:

Or, to get the most out of it:

• Spend your gems only on gold.
  Alternatively, wait until you have 500 Gems and buy the 10,000 Gold pack, and don't buy cards from the store. It is a known fact that you get more cards than gold.
• Never spend your gems on purchasing chests from the store. It's not worth it.
  That is, unless you're trying to get guaranteed Epics or Legendary cards, which you shouldn't be doing until most of your cards are maxed as the cards and gold values don't add up, even without the gem prices scaling with your Arena number.
  One exception to this are the occasional Epic Chest (using gold) and Legendary Chest (500 gems), which are much more profitable than other chests available for gems, or cards for gold.

Nope, your chests are tied to your current - not highest - arena level. If you make it to the top and then fall back to arena 1, you'll get arena 1 chests with arena 1 cards.