I've avoided[1] hunting for so long, because it's not reliable and tends to just put hunters in harm's way. Though it may have been tweeked significntly over the ?decade? since I started to ignore it.
I primarily rely upon crops, for food, so that's not really useful to the thread question, but I take it as a matter of honour to pasture and (where possible) breed every creature I receive (wagon-haulers, any pair I have points to add to embark, any caged creature to come by trader that adds usefully to my stock). I used to cage-trap out in the corners/edges of the map a lot (and pinch-points, as I sprawled ever more defensive barriers), for wildlife and hostiles alike, but the logistics of that changed significantly and I don't do that so much.
For pasturing, I tend to set up 1x5 strips and put both male and female[2] in them, which tends to not overcrowd everyone. For grazers, the strips tend to be straight above my Z-1 subterrainean farms (until and unless I convert that area to Z-1 sunken surface-exposed ones by removing the 'roof' ground) within my walled-off area, and are created ASAP once any trees are removed.
Depending on the grazer, they may need regular moving to the next 'virgin' strip along. I always make sure I have spares (or recovering ones) and there tends to be no normal walking across them for other purposes. If I get a plethora of impossibly hungry grazer-species (elephants were bad at this, but I can't remember how long since I got them other than as wildlife) I get time to arrange further strips, and as a last resort I butcher them all off.
For non-grazers, I do this in 'stables', perhaps rooms dug near the farm-levels (depending on how I progress through the expansion scheme). I underuse "underground grasses" as an option, for some readon. For egg-layers, I tend to revert to one or more standard 5x5 plot with either breeding pairs (and enough nest-boxes to satisfy) or the same-sex 'spares' that I'm not yet slaughtering. The nestbox rooms can be locked to prevent egg-disturbance (prior to making it explicit) whenever other access is not needed.
The young that are born/hatched I tend to leave with the mother for... sentimental reasons..? I'll move the bull onto the 'next' pasture early to reduce the resulting crowding. After some time (or immediately upon notification of maturity) they'll go to the creche/pre-butchery (non-family) holding areas. I may be a little less ethical about the crowding of, say, ducklings (once I've got a high number produced each hatching) but only because I'll be too busy. I keep meaning to have an egg-factory for food (unlocked breeding nest-area, perhaps with my second-best males and females) but mostly I tend to only get 'accidental eggs' in my food-chain.
My butcher's shop, BTW, I tend to put near-surface with an open roof (or sometimes on the surface) with upper access walled-off from invaders and bridge-roof-offable from aerial invaders. That's for anti-miasma purposes, if I can't stockpile/dump[3] enough of its product quickly enough. I try to make this 'chimney' a feature, if it's more than just one level of hole, though I really don't think it adds much (and I
think it just needs to be the central tile, not the full 3x3 footprint).
But I'm more adhered to long-held habits that maybe
once were fine-tuned, rather than an ever-tweaking (and experimenting) perfectionist. My system is far from optimal and may have dangerous flaws in it. That I deal with, perhaps, but in ways I haven't thought to mention right here - being down to a side-issue or general habit of mine not directly related to what I've just decided to describe.
But if it helps to flesh out 'options', on top of all the others already given, you're welcome to this.
It's far too much to solve your immediate problems, but perhaps 'food for thought'...
[1] Actively discouraged. A migrant with a hunting skill enabled (as with fishing) gets it disabled on arrival, but they'll have skills useful for my military (that I also try my hardest not to use in anger, but is there if I need it), and a crossbow! (Ok, not the fisherdorfs... They're effectively haulage-peasants, usually.)
[2] Though I've not established it works this way, I'll butcher the
least-large (or maybe fat) of the males/females of any species, until I'm left with the Prime Pair who
may tend to give larger-biased offspring, and so by animal husbandry I breed my own idea of pedigrees, theoretically selecting for increasing utility in whatever purpose that creature is best put to. This is a holdover from before Gelding and when map-wide spore-breeding was a thing, so I now know I don't need to be quite so hurried to prevent runts from getting involved in this process, upon attaining adulthood. But that's what I tend to eventually do, anyway, just for my own self-satisfaction. It also means that 'pet' runts shouldn't be a problem in despoiling the theoretical gene-pool any more (at one time I'd have perhaps caged them, cages being spore-proof).
[3] My primary refuse/dump-piles are are always in "safe aboveground" spots, for similar reasons.