How often can I expect egg-laying animals to lay eggs

Assuming you want to capture animals (groundhogs, horses, bears, dreaded Rhesus Macauqes) as opposed to vermin.

Build a cage trap where the animals roam (usually next to an outdoor food pile works well) with build, Traps, cage. You will need a mechanism and a cage. Mechanics skill is used to place the trap.

You should probably designate an animal stockpile (p, a) where the animal will be taken after it's caught.

After this, just wait for the animal to be caught.

Note: the capture live land animal option at the kennels catches vermin not animals.

Hint: placing your cage traps randomly can take too long to capture anything. One way to increase the chances of an animal walking into your traps is to channel one long, 1-tile wide, dry moat(d,h) and remove all ramps(d,z) except one in the middle of the moat. Then you surround the single ramp with traps.

Once you've captured an animal, even if you're going to butcher it you probably want to get it trained at least a little, to avoid it breaking free on the way to the butcher shop. To train, make sure you have at least one dwarf with 'animal training' labor enabled, then find the animal in the status tab (z) and designate it for training. Butchering is also handled through the same status menu.

If you just want to fix it (i.e., remove all the piles of vomit) you can download DFHack and use the "clean" subprogram to remove all contaminants on the map. Works on blood, dye, snow, vomit, and ichor, I think.

If you do that and your animals remain nauseous, you might have some underlying cause.