If I am in arena X when I collect my tenth crown for the crown chest, or a free chest becomes unlocked, then proceed to climb to arena X+1 before opening the chests, which arena will be used to determine the cards I get? Will it use the arena I was in when I unlocked the chests in, or the arena I opened them in?
In the case of Free Chests and Crown Chests, the content of the chest will depend on the moment you open it. So it will use your current arena level. This seems to be confirmed in this forum thread :
Next, whatever arena you are at when you receive the chest, that is the level it will stay regardless of what level you are at when it is opened. Crown chests and free chests are based on what arena you are at when you open them.
It also seems to be confirmed in this guide :
The kinds of cards you can get depend on the arena you are in. For example, if you are in arena 3, the chests can contain cards available in arena 3 and below.
Original post is either wrong, or it has since changed. Just tested myself as well. Finished crown chest in frozen peak but didn't open until after I finished my next game (which brought me to 3009 trophies). The contents were that of arena 8 (what I was when I finished the crown chest) and NOT arena 9 (contents were NOT for the arena I was in when I opened it).
Crown chest contents are determined as soon as you get the final crown, (the arena you are in at the end of that match) so you might as well open it immediately.