Clash Royale – Are Clan Chest Rewards Proportional to Contributions?

EDIT: Following the recent update as of 03/05/2016 the current chest cycle can be found HERE

Original Answer: The comments are right, and apparently also is the reddit post.

It is pre-determined when you get a drop by this updated list (to include new chests). I just got a giant, so I know exactly where in the list I am, and can start logging from there. Few things to note:

1. Having no free space, but wining a battle, moves you to the next chest. So In theory you can always get golden and above.

2. The list might get updated or the entire system scrapped by Supercell at any time, given they are the developers and can do whatever they please.
3. Having gone trough the list, it starts over from 9.

A win would push you forward in the cycle, regardless if you have a free slot or not. That's a win where trophies exchange place (eg. not against clan-member in a friendly match)

EDIT: What I've just red on this FAQ is having full chest slots and wining does not progress you on the list

As far as I can tell, no.
The only reason to open the Arena X chests would be that you're out of Arena X+1 chests. Maybe if the higher chest is a gold one and you're saving it for the night or work day.

You many want to clear them out of the way at some point to make room for better ones, but if you're constantly getting just as many chests as you need, I'd postpone opening the lower level ones.

Of coures, they are in your inventory. I'll propably open them at some point anyways. Maybe that time you fall to the lower Arena. Maybe that one day you just don't feel like playing.