I know of people who use pine or newspaper pellet kitty litter and then compost the used litter for TWO years. I haven't tried it, but I am tempted. I do use pine pellets for my cats and chickens. I have never composted dog waste.
I tend to look at it like this- I already know that these materials biodegrade, nature shows us that. The question is why do folks typically recommend that we don't put it in our bins? Rather than accepting a blanket "Don't do it" recommendation, I prefer to look at the concerns with doing it and then see if I can address those concerns in another way. Best I can tell the concerns are the following:
1. bacterial/ other microbe transfer to food and therefore people who eat it. Rather than simply deciding that I won't compost dog poop, meats etc, because it may cause bacteria to grow in the bins, I simply ensure that the bins get really really hot and stay that way for weeks. The bacteria that are a concern for humans don't live very long in temperatures that my compost can reach. For example E Coli seems to be killed/ inactivated in a matter of hours at 140 deg. F. I decided to use the guidelines (a year of mesophilic activity after the ectothermic period is done) in the Humanure Handbook, since they are guidelines for composting HUMAN waste, which I would think is much more worrisome. My dogs are two miniature poodles, and the waste is going into a pretty large bin (about 4 feet x 3 feet x 2.5 feet high) so I think that's quite a small amount in the overall stuff going in. I think that time frame is way on the safe side since that's what they do for whole bins of human waste.. Besides all of this, when people cite disease transmission from fecal matter, they are usually talking about direct application of said matter, and not from the use of composted material. I think it's pretty easy to see if it's composting or not.
2. The risk of bringing pests and predators is often cited for why people don't recommend meats and fats and such. I divide the pests into two categories and answer this in two different ways.
Small pests like insects- My bin is pretty chock full of the black soldier flies (my avatar picture). They actually manage the pest flies, by secreting a hormone that pest flies can 'read' which says essentially- "don't bother laying your eggs here, there will be no food left for your larvae because we eat everything." It's said that their presence can reduce pest fly populations by 90%, and I can attest that I don't really see any at the compost. They love the coop, where I have to hang those drowning traps periodically, but they are never at the bin unless things get really out of whack. These BSF aren't pests themselves because they die after turning into flies and mating, so they won't spread disease by landing on your food, etc. Other critters in the small categorie I see are pill bugs, pincher bugs etc, that don't seem much trouble to me. At the bottom, I have LOTS of worms that are working- and it's cooler down there for them, as the new 'hot' stuff is being added to the top.
Bigger pests- rats, mice, possoms, etc, I manage by building the bin from what my friend calls "paper pallets". These are similar to other pallets, but the space between the slats are barely wide enough for a soldier fly to crawl through. If the occasional mouse gets in some way to munch on things, I haven't seen it yet- though I did have this trouble with my first built bin, as I just didn't make the joints tight enough..
3. Smell! In my experience bad smells in the compost don't have much to do with what I put in, just with how much of each thing I put in. Water- Carbon- Nitrogen. If one of the above is not in good proportion, it will stink whether it has meat or just veg. Keep it aerobic, and have a storage of Carbon materials and it should be fine. When I need a fast influx of Nitrogenous stuff, I go to starbucks and get trashbags full of used grounds.
I also don't turn anything over like most people recommend, until after the hot phase is done. Instead I build it with rougher twiggy stuff on the bottom so that air can come up. And since I'm composting as I go, I actually get many hundreds of little hot phases as I add things. I recommend visiting the bin a bunch (so I take out food waste daily) and just see how things are doing. I used to have one thermometer in the bin, but Starbucks unknowingly donated another like they keep on their steaming pitchers (in a bag of their grounds) so now I have two. I read temps, pull thermometers out, add new food waste, and then stick thermometers back in the pile. This way if things are off, I can remember what went in recently and have a better idea where the balance is off.. When a bin is done with hot phase (springtime) I switch to another (that I just emptied out for the garden) and start adding my stuff there. That bin gets a cool phase till the next spring.
I can't say that this is the best way, but it works for me. And nobody here has been sick with anything out of the ordinary (not much ordinary either) in the past two years that I've been doing it.. As for lemon peels, not only do I put them on, I typically wind up putting on trashbags full at a time, because we juice in bulk (I live amongst lemon orchards) and we freeze lemon juice to make lemonade later. I've never seen any trouble with them breaking down. It could be that the mold on them is penicillin, and that makes trouble for the beneficial bacteria in bins, but our composting happens over a great deal of time anyway, so if it is slowed down, I might never notice..