Here is a test I did with 135 Three of Coins. My method for this was farming the Eye of a Gate Lord mission, and from these numbers, it sure does seem like there is a cooldown for Three of Coins.
1st Set: 6 L
2nd Set: 8 S
3rd Set: 11 G
4th Set: 7 S
5th Set: 14 H
6th Set: 4 L
7th Set: 6 G
8th Set: 20 S
9th Set: 11 G
10th Set: 8 H
11th Set: 4 G
12th Set: 12 P
13th Set: 14 S
14th Set: 10 H
135 Three of Coins / 14 Exotic Engrams = 10.37% Drop Rate
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You should test it with the Black Garden mission's "exotic farm". I did this and I got around 6-7 exotics in an hour using about 35 ToC's. Seems like it is a better farm for me, seeing as I never get exotics with ToC unless I use this farm. I would conduct a test myself but I'm all out of SC and ToC, so that's not an option.
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Nice statistics! Now all we have to do is figure out what that cooldown timer might be. To be as accurate as possible, someone would have to do this: You have 100 coins: Pop one and defeat an Ultra every minute, exactly one minute or as close as you can to every minute 100 coins: 1 per minute = 1 hour and 40 minutes You have 100 coins: Pop one and defeat an Ultra every two minutes, exactly two minutes or as close as you can 100 coins: 1 per 2 minutes = 3 hours and 20 minutes And so on and so on. Once you see the percentage drops leveling off or just about even (ie defeating an eligible Ultra/competing a Crucible game with a Three of Coin active every 7 minutes has the same chance as every 8 minutes, then that's the cooldown timer! I'm not always online to do this sort of thing. So my statistics reflects "I'll-use-them-whenever." Dang. So my cooldown statement got debunked.