“She's had many strange things happen to her recently. = She’s been having a lot of strange things happen to her recently” This sentence is from “Merriam Webster dictionary” about “have” meaning “to experience (something)”. Another example from “Longman dictionary” is “He found it quite natural to have people fussing (not “fuss”) over him.” Why don’t you say “She's had many strange things happening (instead of happen) to her recently.”? like “fussing (not “fuss”)”? Or do you have a grammatical reason or meaning differences between them? And these examples have any grammatical relation to “have something stolen/broken---“ and “My car is having new tyres fitted.” ? And am I right If I think that “have” means “experience”? in the way “have (=experience) something stolen/broken---“ and “My car is having (=experiencing) new tyres fitted.”
The difference is a matter of emphasis.
The first, with the present perfect
... has had many strange things happen to her recently
presents her recent history as involving a number of discrete events.
The second, with the present perfect continuous
... has been having a lot of strange things happen to her
presents her recent experience as a continuum of strange events: things are happening to her seemingly one after the other with little or no opportunity to recover from the event or to assimilate it before yet another strange thing comes along. She has been "beset" with strange events.
Why don’t you say “She's had many strange things happening (instead of happen) to her recently.”? like “fussing (not “fuss”)”?
This is a matter of "mixing tenses" which should be avoided. Have is present; Had is past. "happening" and "fussing" imply ongoing present tense. So "have" would be used with these.
The other examples are similar to the meaning of experience: You can say that you have "experienced" something being stolen or broken; or your car is "experiencing" getting new tires. You just would not normally phrase it that way. Hope that helps.
By the way, although you didn't ask: The difference between “She's had many strange things happen to her recently." AND "She’s been having a lot of strange things happen to her recently” IS THAT "had ... happen" says nothing about it happening again, whereas "been having" suggests that it may be ongoing and continuing.
She's had many strange things happen to her recently.
He found it quite natural to have people fussing over him.
To understand the usage of bare infinitive and the -ing form, let me quote from Michael Swan's Practical English Usage:
"have + objective +infinitive/-ing form.
In this structure, have often means experience. This usage is rather informal.
I had a very strange thing happen to me when I was fourteen.
I looked up and found we had water dripping from the ceiling
Note the difference between the infinitive in the first example (for the thing that happened) and the -ing form in the second sentence (for the thing that was happening)".
I think that satisfies the sentences in question.