As to the farming and storage, do check that your fields and/or storerooms are not filled with hidden objects. (Have you used hide on anything? If so, unhide everything and see if there's already stuff there).
As to farming and farmers not working most of the field - double check those seeds. Are you sure you still have them? Rotting plants leave no seeds. Forbidden or set to be dumped seeds cannot be planted. Forbidden fields won't be worked (or harvested, for that matter).
As to the animals, pastures, and movement. Animal care is only used by the owner of a pet to attempt to heal a hurt pet - however in the current version this is bugged out of service. The skill currently has no known working function, so animal care is meaningless to your fortress.
Both my yaks died with plenty of dense grass around (also on the top half)
That's unusual - pastures are created using i, and (that I know of) can only be one z level. If you are trying to make multi-level pastures, that might be part of the problem right there, as only one z-level of the pasture is part of the pasture (whichever level you actually placed the animals in is the only area they can roam).
Although where they are standing the tile seems to be gray, a sign off my shale rock maybe? But it with "k" it shows Dense satintail/grama.
This seems odd. When you see the animal in the tile, all you can see is the animal - it's not a translucent overlay letting you see the background the animal is standing on. But a creature in a pasture should be eating the grass it is standing on, if hungry (starving to death is hungry indeed). So, are you actually using a pasture? Not a burrow or something else?
As to nests and nesting - when my egg layers are ready, they will outright leave their pastures and head to any unclaimed nestbox they can reach. But it takes time. An animal usually only lays eggs once or twice a season, almost always in the spring and often again in the fall. A new animal (from the caravan or one that just grew up) is rarely ready to lay eggs very soon, and so can miss an season before they first lay.