How does the Piñata contest work? Do we only get one Piñata per day or does it just continue to the next one once the previous one is completed?

you move on to the next one once you bust the previous one open. You open them once you get enough points on them and you fulfill the points bar by clicking on them. We move through the first couple pinatas pretty quickly because the points required are so low but once you start getting to the later ones it's possible to be on the same pinata for 3 days or so.
It's possible to increase the number of points you get per click by completing objectives and you can also decrease the time in between clicks by doing objectives as well, i.e go from once every 8 hours to clicking every 6 hours, etc.

Basically you have 20 pinatas. Each pinata has a certain amount of points you collect (for example, it might 100 points). You get a certain amount from the daily clicks and daily objectives. There may be sometimes that the objectives say the next click opens the pinata, or a click equals X amounts of points. Once you finish a pinata, you get the prizes and can immediately start on the next pinata. Now, a pinata may take more than a day to open. Usually the first couple are pretty easy to open because they are low amounts of points needed. Does that explain it well enough?

the new divine series "horses of the vegetable kingdom" (that's what that translates to at any rate) that we will see in the Pinata promo. You can see it in this post here.

Yggdrasil, the new divine supposed to replace pass horses, the "broccoli horse" because it looks like broccoli and is pretty useless

Vegetable is so confusing,,
I'd rather call that "plant kingdom" , well that's maybe just me,,

What does the "Broccoli Divine" look like and what does it's "bonus/Special" page say its supposed to do?

pinatas and snowball fight

other than grow and have its image change over time as you send older horses to Heaven. A photo album appears on its page you can scroll throw to see its image changes. You can't care for it in any way, and aside from the photo album the only action you can do on its page is add the tree rings to it, which you get by sending old horses to Heaven.
This is YggdrasilThat one is mine and I don't have a photo album on it yet because it's not old enough I think.

As for the perks, it has none. Literally the only thing it does is grow when aged with tree rings that you will be able to get when horses older than 25 die. It has no skills and cannot be cared for. As its image changes, you will be able to view a photo album of its previous appearances, and there will also be a ranking for the oldest Yggdrasil.