Each year this question gets nibbled on (and around) in the spring as compost bins get opened and spread on gardens. Below are my take on my compost needs. YMMV
Q., Do you need utterly decomposed and homogenous compost?
A,. No, but there is a chunk size I find inadaquate to my need. If I find a leaf stem or twig its probably going direct on the new bed, and getting turned in.
There are woody bits like brussel sprout stems that are just too brushy to go to field, and will get tossed back on the pile for another at-bat.
Q,. Do I need to screen my compost before using?
A,. I do screen some of my compost that will go into soiless mix I make up. But I suspect this has more to do with my esthetic, than my plants needs.
Q,. At what point will critters leave my compost alone?
A,. Probably never. Mostly its not the kitchen scraps that visiting fox, voles, mice, dogs, possums are after in your compost. it is the indigenous micro herd that they fancy as snacks.
My solution has ever been to enlarge the buffet, rather than to limit access. I will use a closed can to decompose kitchen scraps by using a steel can*, before adding them to compost bin. I always put some finished compost out on the lawn.
After many years of observation only bear have the potential to endanger humans. Everybody else is looking for a clump of earth worms for a late night snack. And prefer their worms where they can watch the neighborhood.
By precomposting kitchen scraps it mostly liquifies and adds nicely to the center mass of your compost bin. I started doing this to get a neighbors dog out of my compost bin. I got tired shoveling up the second-hand compost he ate.
It did not matter if I added or withheld food items from the compost, Ferdinand the great dane would try to eat his own body mass of compost till I started precomposting kitchen scraps.