Yes, they can take reactions
The last paragraph outlining the companion (added in an errata after the first printing).
Like any creature, the beast can spend Hit Dice during a short rest. If you are incapacitated or absent, the beast acts on its own, focusing on protecting you and itself. It never requires your command to use its reaction, such as when making an opportunity attack.
This tells us three things.
- Animal Companions get a Reaction
- Animal Companions can take Opportunity Attacks (using their Reaction)
- You don't have to expend any command (i.e. - an action) to cause/allow your companion to make an Opportunity Attack as a Reaction
Crawford's tweet seems to indicate that he agrees, but I wish he'd been more direct about it.
Maybe a Bonus Action (probably)
As you mentioned, there's not normally a situation that would grant a companion a valid bonus action. They can't two-weapon fight or cast quickened spells. I can't definitively say "There is absolutely no situation that would grant an Animal Companion a Bonus Action", so I won't.
In the case of the Panther's Pounce special quality, I don't know. It's something I'd have to see in play to make a ruling on. The RAF and RACS answers seem to be that the animal companion gets to use its bonus action, like any other creature. But what about RAI and RAW?