british english - Have got in BrE

I am really confused now, so I've found out that:

Facts :

HAVE GOT is only used in the present and past form.

the plain "have" is used in all tenses.

HAVE GOT cannot be combined with the modal verbs such as :must, may, might, can, etc. and can't be combined with to+infinitive.

Only the plain "have" can be combined with the modal verbs & it can be combined with to+infinitive. ex. I want to have a pair of shoes (or) I want to have got a pair of shoes.

HAVE GOT and the plain "have" are used in BrE. (Do BrE natives use both ? Could you explain the differences between them?)

The pattern

A. HAVE GOT : I have got.. & I haven't got..& have I got (in the negative, haven't I got ___ ?)

B. HAVE : I do have.. / I have.. & I don't have.. & do I have? (negative, don't I have?)

C. past : I had got /(I did have) I had & I hadn't got /I didn't have & had I got (hadn't I got?), did I have? ( didn't I have?)

Are these facts correct? I await your explanations, thanks.