Most of my grazers are intelligent enough to go on they own when assigned a pasture. There's even no need for a dwarf to go and get them. Even cats walk on their own. Some of animals (not only cats, but also grazers) need to be hauled sometimes. I have a checkerboard pattern outside, now used only for war elephants, and when I assign one of them to another pen they immediately go there. The same goes for most donkeys, cows, llamas etc. I think this may be partially individual, not species specific. Or random.
That's... weird. Are you defining a meeting area in the same zone as the pasture? That's the only way I could see animals deliberately walking toward some form of confinement.
I avoid grazing animals, so most of my pasture use is for keeping the five species of birds I keep separate.
Hm, no. My meeting areas don't overlap with my pastures. I use version .25 and only thing I changed regarding animals was removing _EXOTIC from PET_EXOTIC tag.
I have a specific pen area for birds, it covers 10 nestboxes, and couple of chains (for males, to limit their fights), and only see a bird moving to adjacent door occasionally. There are exceptions - newly hatched chicks need to be hauled to a cage.
When I order 10 of my animals to go, usually 8 of them go on their own, and 2 need to be hauled. I don't know if they really need it, maybe if given enough time they would go on their own. There are jobs created (Pen/Pasture Large Animal), but most of them disappear before dwarves come. If a dwarf got assigned to move animal, but it managed to move on its own before dwarf did it, the dwarf will come, take the animal, move it one square and go to drink or something.
EDIT: Oh, and I changed the amount of feed elephants need (to yak level), but it doesn't matter when they are trained as war animals or taken as pets, they don't seem to eat then.