If 2 unmarried parents live together, share 3 kids & split the bills evenly, can 1 parent claim 1, the other parent claim the other 2, and each claim head of household?

Only one can file as head of household. The other files as single. 

No, you can only have one head of household, as that individual needs to furnish over 1/2 the cost of the home. You can choose who claims the children as their dependent, as long as they qualify otherwise.

[Edited 1/23/25 at 3:25 PST]

@Omarlrobles