An animal may be led to graze in a grassy area, and it does not constitute Kotzer since the animal is eating for its own sake. We are not commanded to make sure that animals keep Shabbat, only that they do not labor for our benefit (Shabbat 122:1; SA 324:13).
One may provide food and water to animals in his possession that depend on him, like cows, chickens, and cats. Food and water may also be provided to animals belonging to another Jew. However, one may not feed or water self-sufficient animals like bees and doves. Even though feeding them is not a melakha, the Sages prohibited it because it is seen as requiring excessive effort (SA 324:11; BHL s.v. “ve-yonei”).
One may put food out for hungry animals like stray dogs and cats. As we know, God shows mercy to all His creatures, as it is written: “His mercy is upon all His works” (Tehilim 145:9). It is proper to emulate His ways (AHS 324:2-3; see MB ad loc. 31).
Animals are muktzeh. Therefore, they may not be picked up, nor may their limbs be lifted up. However, when it is necessary for their well-being (for example, in order to get them to their food), one may take hold of them and move their limbs to prevent them from suffering (tza’ar ba’alei ĥayim), as long as one does not lift them off the ground (SA 308:39-40; MB ad loc. 151).
If animals have difficulty eating unless the food is placed directly in their mouths, one may feed them in this way as long as he does not force-feed them. Force-feeding refers to shoving the food so far down their throats that they are unable to spit it up. This requires excessive effort (Shabbat 155b; SA 324:9-10).
One may ask a non-Jew to feed geese that have been force-fed for so long that they are unable to eat otherwise and would suffer hunger and pain if they were not fed on Shabbat. In such a case, one may ask a non-Jew to feed them one time on Shabbat. If there is no non-Jew available, the poskim disagree whether a Jew may feed them to minimize their suffering (MB 324:27). It is better not to force-feed geese at all, as doing so causes them pain and entails multiple prohibitions (see BHL ad loc.; SSK 27:26).
One may cut up food for animals if they would not be able to readily eat it otherwise. This includes cutting up pumpkins that are fed to animals and tough carcasses that are fed to dogs and that they have trouble eating. However, one may not cut up any food for them that they could eat by themselves. Even if cutting makes it easier for them to eat, it is excessive effort and thus forbidden on Shabbat (SA 324:3-8).