what are they specifically? how do they work and how do we stat nurture? how much do stats increase while nurturing how so many times? which options increase which stat?

"Stat nurtures" is a term for nurturing hatchlings in order to raise a stat. They're different than nurturing eggs, which only makes the egg hatch faster with no effect on stats.

Here are the proper actions to take:
Hungry --> Offer it food
Slow --> Race with it
Derpy --> Read a book to it
Weak --> Work out with it
Restless --> Play with it.

When a pet has been nurtured five times (as a hatchling - if the pet was nurtured 11 times as an eggling, then the first stat point would be allocated at 16 nurtures), the corresponding base stat will increase by one (or ten for health).

Feeding: health
Racing: agility
Reading: intelligence
Working out: strength
Playing: dexterity

BUT you /can't/ increase a stat if the nurture prompt doesn't ask for it. Choosing the wrong option will just waste your nurture for that day.

Typically it prompts help for a pet's lowest stat, up until it hits the maximum (2000 base for health; 100 base for all other stats). So if a pet has 140-12-11-10-11, it will start with the 10 stat. If it gets to 140-12-12-12-12, it will keep cycling through from the top to the bottom until the pet grows up. Health is a bit of a wild card, but it seems to count similarly... I think 140 from the example would count as 14 in the cycle? I'm not sure how much "research" has been done on health boosting though. A lot of the time health seems to be the last priority, so it's common to see almost-max-stat pets with 100s in every other stat but still 1800 or whatever in health.

So far as I've worked out, it seems like HP goes by the middle two numbers of the four possible?

So a pet with 150 HP, and 15 in all the other stats will want food, but HP jumps by 20 when it's boosted, so it'd go up to 170 HP and not boost again until all the other stats hit 17.

Not entirely sure how it goes for pets with 1000-1080 HP? I'd assume pets with those numbers would always ask for food?
2000 HP pets always ask for food from what I've seen though.