Yes, ISS is connected to NASA through its programs.
ISS
they didn't.
NASA
ESA, CSA, Roscosmos and JAXA
Go to the related link - a java based program from Nasa to tell where certain satellites including the ISS are.
It isn't a NASA space station but the InternationalSpace Station or the ISS
NASA will buy seats on Russian launches.
No single person designs a space station, or any large and complex machine nowadays. A NASA space station would be designed by a team commissioned by NASA. The ISS is an international undertaking.
ESA is the European Space Agency, which has been involved in the building of the International Space Station along with NASA and some other countries.
I mean in microgravity
You've answered the question already. It is called, "International Space Station", or "ISS" for short. NASA uses the name, "Alpha" to identify it, but the other ISS partners do not.