1. The raking idea sounds best. We water our lawn each morning when we can and it just disappears. The reason for the tractor is so the chickeys aren't walking around in their poo the next day. If this is a highly manicured backyard, then the rake, used upside down should handle any big poos and break it up. Water when needed, but not just to dissolve the poo. it will break down pretty quick. 
2. If you have a dog and worry the dog might ingest poo--mine does, so I make sure it is broken up and spread away--then if you pick up that poo where you see it, pick up the chickey poo where you see it. BUT
3. I would consider a compost pile to put your chickey poo in (not dog poo) also kitchen scraps-nonmeat--leaves, dead plants, old potting soil, etc. Makes great mulch for use around trees, plants, etc. not much work either. When you add some chickey poo to the pile, just use the rake to turn it good. done!
4. There's all kinds of things people worry over that have little impact to their lives. I think what you are tracking inside from your yard is one of them. Too much attention to de-germing can cause the opposite reaction--being more susceptible to disease, not less. Your body needs to 'play' with germs you contact to keep your body ready for fighting duty. If they don't have 'enemies' to fight, they get lazy and won't be up to duty when needed. If you are worried about germs cause baby crawls on the floor, don't. Babies need contact with the real world or else they will develop allergies and have a harder time with colds, etc-- for the same reason I mentioned above.
5. Simple solution for the tracking in thing is simply to take your shoes off at the door. Lots of households have adopted this Hawaiian habit and enjoy it. 
--We'd love to see pictures of your tractor and chickeys! Hope this will be helpful.