You did notice that something is different between American and British usage of "have on a hat". Let me show two Google Ngrams that illustrate the situation.
British:
American:
In British English, pre-1960, "had on a coat" is the most frequent of these expressions, followed by "had his coat on". However, "had on a coat" drops in frequency by a factor of nearly 30 by the 2000s.
In American English, these same two phrases are also the most common, but remain so for the entire period.
(Don't ask me why the most likely order of the preposition and the object depends on whether you use "a" or "his"; this is a piece of grammar that I obey, but I have no idea why. But either order is grammatical with either "a" or "his" in the U.S.)
To answer some of your questions:
(1) I think Americans are roughly equally likely to say "had a black hat on" and "had on a black hat". However, you are only noticing the second one, because this is a usage not found much in England anymore.
(2) If something was grammatical 50 years ago, I think it's a stretch to say that it is actually incorrect today. But you are right in that "had on a black hat" seems to be falling out of use in England.
What is going on here? To have on meaning to wear is a phrasal verb (Wikipedia link) made up of a verb combined with a preposition1. For many phrasal verbs, the object can go either before or after the preposition. For some, like to tell apart, the object can only go before the preposition. For others, like to take after, the object must go after the preposition. What I suspect is happening is that, in England, to have on has changed (at least among the younger people) from the first kind of phrasal verb to the second.
1 This preposition is sometimes called a particle because it does not behave grammatically like a preposition when it is part of a phrasal verb.
AmEng speakers do NOT use definite articles all the time; whether we do or not depends on what we want to say, and how we want to sound saying it. In particular, the use of the definite article with the present tense of the verb "to be" depends (in the words of a famous American) on what your definition of "is" is. Several of your example sentences sound distinctly unnatural to my USAite ear. Here are my suggestions based on my own usage:
School:
- He is in school. - He is a student; specific time does not matter. Perhaps it's Friday night and he's partying right now.
- He is in the school. - This does not sound natural at all to an American ear.
- He is enrolled in the school. - He is a student.
- He is at school. - He might be a student OR a teacher, but he is on the premises at this moment.
- He is at the school. - "The school" is a local landmark, and he's there right now.
Hospital:
He is in hospital. - He's an inpatient. This is valid, but not usual American usage. Anglophiles (Americans who like to drop Briticisms into their speech to appear more sophisticated) will sometimes use this.
He is in the hospital. - He's an inpatient.
He is at hospital. - We don't say this.
He is at the hospital. - He might be an outpatient, or he might work there; either way, he's there right now.
Church:
He is in church. - Services are in progress right now, and he's there.
He is at church. - Interchangeable with "in church".
He is in the church. - He's inside the building; no information is conveyed about what he's doing there (he might be polishing the floor, for example.)
He is at the church. - He's on the church grounds, not necessarily inside the sanctuary.
University/ College:
He is in university. - Americans don't usually say "in university"; "in college", however, means that he's a student.
He is enrolled in university. - He's a student.
He is at university. - He's a student. This is much more common than "in university", for some reason.
He is in the university. - Again, we don't say this.
He is at the university. - The university is a local landmark and he's there now. (Who is he - student, professor, tourist? Not enough information.)
Prison:
He is in prison. - He's a prisoner.
He is at prison. - We don't say this.
He is in the prison. - He's inside the building. He might be a prisoner, a warden, a visitor...
He is at the prison. - He's on the grounds, not necessarily inside the building - he might be waiting in the car while his wife visits her brother.
Best Answer
I have to admit I haven't consulted any sources, but here is what I'd say is "correct":
and
and
"Weekend" would not normally be pluralized when preceeded by the word "the". For example:
And finally, to address "at the weekend": this is often seen in British publications, but I've never seen the phrase "at the weekend" in American English.