There are many cooldowns in this game, and many activities.
Your wolf can be rendered ‘Untradable’ during the time it is on these cooldowns (from what I can tell):
- Playing with companion cooldown.
- Removing from mining cooldown.
- Removal from Gauntlet cooldown
- Removal from Explore party cooldown.
I cannot see anywhere on my wolves' pages a place that indicates these cooldowns. There really should be indicators directly on the wolf what these cooldowns are and when they will expire, so that you don't have to search every activity page in the game to see where your wolf is occupied.
And the companion cooldown is just confusing, I don't see any indicator for this at all. All I know is I cannot remove the companion from a wolf I wish to trade, and therefore cannot trade them. And I believe it is connected to the Play timer.