I saw a shared video of someone with a buddy with catch assist. It seems like a normally wasted ball (landed during the 'attack' of the wild Pokémon) is being headbutted back, delivering a 'Great throw' (I personally think it would be an awesome combo). Are the normal catch chances applied? Is this assist always there? Does the original circle matter for the type of throw, or would it always be great?
Are the normal catch chances applied?
Yes, the normal catch chances still apply. Pokémon can still break free if your buddy throws the pokéball back, so it's safe to assume normal catch rates still apply (source needed).
Is this assist always there?
No, this does not always apply. Much like Critical Captures in Pokémon Go, it is a random chance for Catch Assists to trigger.
Does the original circle matter for the type of throw, or would it always be great?
From what I have observed, your buddy will kick the pokéball back to the center of the catch circle. Some videos show catches that were well outside the catch circle that would have not awarded a catch bonus did award the respective bonus based on the circle size.
Curve ball bonus' also still stick! If you apply a curve throw, miss, and your buddy throws the pokéball back, you will be awarded the circle ball XP bonus if the Pokémon is caught