First, "Everything has two handles" says that all-that-is, or the universe, has two handles. The sentence is grammatically correct but not highly meaningful. The variant "Every thing has two handles" also is grammatically correct and more or less equivalent in meaning to "All things have two handles" except that using Every avoids the ambiguity (noted below) when All is used.
Second, while "All men have one head" is grammatically correct, it is semantically ambiguous because of ellipsis. Most people will take it to mean "All men have one head each", but as noted in the question, it could be taken to mean "All men have one (shared) head". Mostly-unambiguous rephrasings with various meaning and nuance include
Every man has one head.
Every man has his head.
Each man has one head.
All men have one head each.
All men share one head.
The phrasings "All men have a head" and "Every man has a head" are grammatical but like "All men have one head" ambiguous, as head could mean either one head each or one shared head. "All men have heads" would usually be taken to mean "Each man has his head" but does not preclude people having multiple heads.