(Some of this is drawn from https://www.halachipedia.com/index.php?title=Animals_on_Shabbat)
I think Sefaria's language of "Sabbath-observant animal" was a bit confusing.
Any animal owned by a Jew may not be prompted to do activity prohibited for a Jew themselves to do on Sabbath.
Let's break that statement down, and hopefully all your questions will be answered. The key verse on this is Exodus 23:12; so that your ox and donkey can rest like you.
Maimonides Sabbath 20:1.
אסור להוציא משא על הבהמה בשבת, שנאמר "למען ינוח, שורך וחמורך" (שמות כג,יב)--אחד שור ואחד חמור, ואחד כל בהמה וחיה ועוף.
One may not have an animal carry out a burden on the Sabbath, as it says so that your ox and donkey shall rest -- this applies whether it's an ox, a donkey, or any domesticated animal, wild animal, or even bird.
Can't be prompted
Rashi on Exodus 23:12 (quoting Mechilta):
למען ינוח שורך וחמרך. תן לו ניח, להתיר שיהא תולש ואוכל עשבים מן הקרקע; או אינו אלא יחבשנו בתוך הבית? אמרת אין זה ניח אלא צער:
Give your animal a rest -- but that allows the animal itself to detach grass from the earth and eat it [though you couldn't use it to power your own reaping]. Does "let your animals rest" instead mean "lock your animal inside?" No; that's not rest for the animal; it's painful.
So that's where the "Sabbath-observant" language got confusing. My cow is welcome to go do whatever it wants to do. I just can't work it.
A lot to unpack there. Let's start with "carrying in a public space." I'm allowed to wear clothes in an unenclosed public space (for modesty, decoration, or protection from the elements) but not carry a backpack. Same thing -- I can put a sweater on my animal if it's cold, but not have my animal carry a burden for me. I could hire a Sabbath-observant improv troupe to perform at my event as there's no melacha per se involved (there are some additional conditions on that, but that's for another time), so I can ask my dog to entertain folks by rolling over.
The case in the Mishna concerned oxen, which everyone knew were used to carry loads in the public thoroughfare, plow, or otherwise do Sabbath-prohibited labor.
Let's get to your questions, then:
are we under any obligation to teach them Sabbath observance?
It's not about teaching the animal. It can do as it pleases. If it's a day when you can't work, you can't tell it to work. (It's not just Sabbaths, it's holidays too by the way.)
"Could a pig become Sabbath observant?" (I'm thinking of pigs trained to look for mushrooms and the like)
We're generally not sanguine about Jews raising pigs, but the point is the same. I suppose sniffing per se is not prohibited labor (I could hire a Sabbath-observant chef to sniff my salad and tell me what it needs), but the digging/detaching of the mushroom would be.
I know these rules were originally set when animals "working" was typically limited to farm animals like oxen who plow fields. But now we have animals trained for myriads of work.
The key here is Sabbath-prohibited labor.
Dogs are trained to work for the blind, or to comfort those that are sick, or to shepherd sheep, etc.
Therapy's the easy one, as there's no "Sabbath labor" involved. If you can cuddle with your own dog, you could have your dog provide therapy for others.
Service dogs: let's do an easy one -- a dog smells indications of an epileptic seizure and warns the human. Sniffing is allowed on Sabbath. Moving on to the seeing-eye dog: if the neighborhood has an eruv enclosure and carrying is permitted, then you're fine. If not, this gets slightly trickier. I can walk my dog on a leash because the leash benefits the dog (like when I wear a raincoat); it keeps the dog from running away where it could be hurt. The harness on a seeing-eye dog could, however, be viewed as entirely there to help the human and thus more akin to an ox carrying my burden. (Shemirat Shabbath Kehilchata Ch. 18, footnote 62. Rabbi Auerbach allowed it, though his student Rabbi Neuwirth isn't so sure.)
Herding sheep -- anything you'd be ordering the dog to do? Once it's trained to herd, you don't have to stop it from doing what's normal.
Bomb-sniffing rodents -- obviously, "save a life by stopping bombs" comes above all else. But even without that -- sniffing isn't Sabbath-prohibited labor.
And is a Jew required to teach any animals that work for him to observe the Sabbath?
Your obligation is entirely passive. No obligation to "Sabbath-train" your animal. (There is a fun story about some holy rabbi's animal that refused to work on Sabbath, even when stolen by a non-Jew; but that's not what the law requires.)