Clash Royale – How Trophy Amounts from Battles Are Determined

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

Epic cards, like Prince, can only be requested on Sundays.

On Epic Sundays, you are allowed to request Epic cards once. The number of Epics you can request at one time is equal to the number of Rares you can request.

Legendaries cannot be requested.

You will receive 1 XP and 5 Gold for each Common card donated or 10 XP and 50 Gold for each Rare card donated. For every Epic card donated, you get 10 XP and 500 Gold.

Source: Clash Royale wiki, "Clans - Donations and Requests"

Here's the full mechanics on card requests / donations:

Clan Members can request Common, or Rare cards once every 7 hours with Epics available on Sunday. Cards received depend on your Arena level and card class:

Goblin Stadium
Donations – 1 Common or 1 Rare
Requests – 10 Commons or 1 Rare

Bone Pit
Donations – 2 Commons or 1 Rare
Requests - 10 Commons or 1 Rare

Barbarian Bowl
Donations – 2 Commons or 1 Rare
Requests – 10 Commons or 1 Rare

P.E.K.K.A.’s Playhouse
Donations – 4 Commons or 1 Rare
Requests – 20 Commons or 2 Rares

Spell Valley
Donations – 4 Commons or 1 Rare
Requests – 20 commons or 2 Rares

Builder’s Workshop
Donations – 4 Commons or 1 Rare
Requests – 20 Commons or 2 Rares

Royal Arena
Donations – 6 Commons or 1 Rare
Requests – 30 Commons or 3 Rares

Frozen Peak
Donations – 6 Commons or 1 Rare
Requests – 30 Commons or 3 Rares

Jungle Arena
Donations – 6 Commons or 1 Rare
Requests – 30 Commons or 3 Rares

Hog Mountain
Donations – 8 Commons or 1 Rare
Requests – 40 Commons or 4 Rares.

Electro Valley
Donations – 8 Commons or 1 Rare
Requests – 40 Commons or 4 Rares.

Legendary Arena
Donations – 8 Commons or 1 Rare
Requests – 40 Commons or 4 Rares.

You cannot receive entirely new cards by request. It’s necessary to first discover them through Chests.

Donating cards also gives you Gold and XP:
Common > +5 Gold & 1 XP
Rare > +50 Gold & 5 XP

Source: Supercell Support's Card Request / Donation article