>Medviediev met WITH Obama at the airport - znaczy, że Medviediev z
>Obamą zamkęnli się w jakiejś sali na lotnisku i mieli tam jakieś
>(ważne) spotkanie. ?
They said hi and talked about something. Perhaps they walked around the airport.
>Medviediev didn't meet WITH Obama before July 7 - nie siedli razem i
>nie rozmawiali przed 7 lipca. ?
No, the didn't.
>Dzisiaj spotkałam Sally (idąc do domu) - I met Sally today.
Yes.
>Jutro prawdopodobnie spotkam kilku przyjaciół - I'll probably meet
>some of my friends tomorrow.
Hmm. You'll probably see them. Or you will meet some new friends.
>W poniedziałek spotykamy się u Johna z naszą starą klasą (robimy
>spotkanie klasowe) - We're going to meet with our old class mates at
>John's on Monday.
To reminisce? Yes.
>Po południu spotykam się z lekarzem - I'm meeting with the doctor in
>the afternoon.
To discuss your case? Yes.
Say "meet a person" unless you could as well say "have a meeting/discussion...with" or "join a person for consultation/conference...":
I met with the boss for an hour yesterday.
"I met the boss for an hour yesterday" is odd.
On top of that, use "meet with" before abstractions when you mean "encounter, experience":
They met with resistance, delays, difficulties, oppositions...
Forget the English vs. American nonsense.