It is absolutely normal for some of the teeth to fall out of a skull you are processing. The gums that once held them in place are now gone. It is also normal for the mandible to split into two pieces in many animals, like this coyote.
Take photos of the skull/head with teeth in place before starting to clean it if you think you will have trouble putting them back into the correct place.
Pictured above is what you do NOT want to do! I received this absolutely glue saturated opossum skull from someone I though would know better. Second photo shows the same glue (white) before I brushed it off with a toothbrush, after I soaked the whole skull in water to remove the glue. Not all glues will come off in water but I suggest that as your first try.
Followup with a peroxide bath overnight. Then dry. Next you put the teeth back into place one by one with a very small amount of white school glue on each root before inserting into the skull. Wipe off any excess glue or it will be shiny on your skull. Let dry upside down so the teeth stay in place.
You put the mandibles back together with a small about of white school glue as well. They will not stay in place while drying without some help. I have a blog post already on how to reattach a mandible.
Some animal skulls you can flip over and the freshly lightly glued mandible just stays nicely in place while drying. Also, by doing it that way you're sure you have them glued together at the right angle.