It is 11:30 in the morning and someone visited me from 10:00 to 11:00 the same morning. Can I say : she has visited me this morning, even though the visit has finished, but it is still morning, or should I say she visited me this morning, or is this only possible when the morning has finished=after 12:00 o'clock? Or are both possible?
You can say "She has visited me this morning" because it's still morning. And you can say "She visited me this morning" if you are sure she'll not come again until noon or if you feel like the morning is over (and you consider it to be past although technically it's not afternoon yet). That is, both sentences are correct but the use of the Present Perfect or the Past Simple makes the meaning a little different.
Here is what Advanced Language Practice by Michael Vince says about the choice between the 2 tenses:
Choice between past simple and present perfect for recent events may depend on the attitude of the speaker. This in turn may depend on whether the speaker feels distant in time or place from the event.
I've left my wallet in the car. I'm going back to get it.
Here the speaker may be about to return, and feels that the event is connected with the present.
I left my wallet in the car. I'm going back to get it.
The speaker may feel separated in time from the event, or be further away.
So, if you feel like her visit is all in the past, you can use the Past Simple.