Why do Grignard reagents form? Why does magnesium insert itself between a carbon atom and halogen atom? Can anyone give any vague idea about why this occurs?

Also does a Grignard reagent only form when there is carbon atom and halogen atom with single bond between them? For example, can magnesium insert between the carbon and oxygen atoms of a carbonyl?
Yes, this is a very interesting question that till today, not everyone agrees the precise mechanism as to how it works. Here is one with homolytic fission:



For your second query, I don't think it is feasible as the $\ce{C=O}$ carbonyl bond is rather stable.