These are really possessive phrases. The only prepositional phrase it the one about your hair.
Neither today's morning or morning of today is correct. It would just be 'this morning'. Likewise it is 'yesterday morning', 'tomorrow morning'. Why? Probably just idioms.
'the room's ceiling' is fine. The later example would need to be 'ceilings of the rooms' or 'ceiling of the room'.
When speaking about you hair and your head, you use a preposition rather than the possessive form. If you're talking about hair on other parts of the body you use an adjective or a noun. As in 'armpit hair' or 'pubic hair'.
Instead of either example you give use 'hair on your head', by default it is understood that if you are talking about hair you mean the hair on your head.
The people's mind examples are wrong because the nouns both need to be in the plural so it would be "people's minds" or "minds of people".
The Earth & sees example has a misspelling. It is "Earth's seas" or "seas of Earth".