Yes, they still exist! In my limited experience the milk does taste better but I think it's because of management, not genetics.

The "get big or get out" Holsteins of conventional dairying will produce themselves to death. That is, they'll make milk even at the expense of their own bodies. To handle that, they need to be fed very intensive grain-based diets (corn and soy). Otherwise they'd waste away.

But low-input cows can produce milk on grass-based diets without wasting away. Grass milk is healthier (https://extension.umn.edu/pasture-based-dairy/grass-fed-cows...) and in my experience, tastier, too!

One of the local dairies has better milk, and anything derived from milk, than the others. Their secret is that some of the herd is of Guernsey and Jersey breed. Milk from only those cows is yellowish in hue, and certainly richer in fat, some would say it tastes better but it definitely tastes different. I prefer the combined milk from the dairy in question (in the US) to pure Guernsey, which is also available, depending on locale.

I would say that the effect of breeding for quantity is less extreme for dairy than for chicken meat, but it's there. The main impact on flavor isn't actually higher production by the cows, it's large-scale homogenization and high-temperature pasteurization as side effects of scale. The larger processing machines oxidize the milk more, and longer pasteurization increases shelf life but it does reduce the taste/quality further. I'm not talking about UHT, which produces a quite noticeable difference in the milk, one which some people like, just processing which leans more in that direction.

Yes, but because of the way that the milk industry is shaped in the US it’s exceedingly hard to get and many people are afraid of it.