grammar - "of the" vs noun adjunct

Please note: This may be a complex question, references would be great, search engines do not help with "of the".

Looks like we can remove the use of "of the" with a noun adjunct switching the order of a couple of names, at least sometimes. Examples:

1. The name of the hotel → The hotel name


2. The handle of the teapot → The teapot handle


3. The use of the "of the" digram → The "of the" digram use


4. The use of the digram "of the" → The digram "of the" use (?)


5. The dog of the house → The house dog (?)

Looks like sometimes (when?) we need the use of the Saxon genitive. Examples:

1. The dog of John → John's dog.


2. The bag of the child → child's bag


3. The beginning of the day → the day's beginning (?)


4. The mistakes of the people → people's mistakes (?)

The questions are:

1. First and foremost: is this correct? (or "what is incorrect here?")


2. Which use is preferred? Using "of the" or the noun adjunct.


3. What rules apply in this context?


4. There may be some cases when the only correct option is using "of the", are there such cases?

Answer

Most of your examples are 'correct'. However, colloquial, idiomatic English tends to avoid 'little' words. More often than not 'of the' is avoided. So the teapot handle rather than the handle of the teapot. This is especially the case in spoken English.

Although not necessarily the case, and easily over-ridden in spoken English, 'of the' can be used to convey formality and also for emphasis, (because word order guides understanding and the first noun in a string is assumed to be the dominant one).

The name of the hotel (emphasising name)

The beginning of the day (emphasising beginning)

The Name of the Rose (formal, poetic)