The topic of (certain) animal waste in compost piles is controversial. However, almost all sources/experts agree that the manure of herbivores is
not only allowable, but positively *desireable
* in compost.
So manure from cows, horses, sheep, goats, llamas, rabbits, guinea pigs, hamsters, chickens (if you can stand it...) is good stuff for the compost heap. (I may have inadvertently omitted an herbivore or two: for example, if the circus comes to town, elephant dung would be good, too, but most of us are unlikely to have continuing access to such!
It's the dog/cat question which raises argument, both for and against.
I do not use either in my compost, although I live with both dogs and cats. But this is due more to my compost being a slow / "cold" operation than having any qualms, whether aesthetic or health-related, about dog poo in the compost. If I were running a HOT compost heap, I'd probably at least test-compost the dog poo. My own dogs are healthy and do not have parasites. They take no meds which would affect the worms/critters in the compost pile (no, they are not on monthly heartworm preventive; I have them tested once a year via blood test per my vet's advice).
OTOH, if there were a litter of puppies here who had just been wormed, it would be a really *bad* idea to put such waste into the compost pile because of the effect it would have on my composting worms.
But even if I *did* have a HOT compost heap, I personally would draw the line at cat waste. Cats being carnivores, their intestinal flora are completely different from human / dog waste. Cats can also be asymptomatic carriers of toxoplasmosis, and I certainly don't want to give the cysts of this disease *anywhere* to grow.
N.B.: There was a die-off of California sea otters four or five years ago which was ultimately tracked to flushable cat litter. Evidently, treatment at sewage plants was insufficient to kill all the toxo cysts, and the sewage effluent--legally "clean" stuff!--carried the cysts out to the Pacific Ocean off the Monterey coast, where the otters acquired the disease. If even modern sewage-treatment facilities can't kill toxo, my compost pile stands no chance at all.
You will find "experts" all over the map on the question of dog waste, but there a large majority of them discourage the use of cat waste in compost piles. Especially after the results of the otter study.
W/regard to your weed seeds: again, a hot compost pile will kill weed seeds. A cold one (like mine, for example) will not. So my pulled weeds--when I've messed up they've gone to seed--go into my yard-waste container and not my compost pile. I do put weeds into the compost when they're young and leafy, though, and they disappear most satisfyingly!
I hope this gives you some useable response to your questions; if not, ask again!
And welcome to The Helpful Gardener, if I haven't already said that elsewhere.
Sunset Zone 17, USDA Zone 9