From the dialogue between Matt and Ryan we learn the following:

1) They don't know each other well - which indicates Dr. Stone is not a career astronaut. It's a small office. Those people know each other.

2) Dr. Stone normally works at a hospital in Illinois. Matt refers to her working 18 hour shifts. This implies she is a normal practicing doctor, probably one that does overnight shifts in an ER. And again, it indicates she is not a career astronaut, because if she were, she would be living in the Houston area.

3) The hardware she is installing in Hubble is "hers" and is based on hardware she has designed for hospital imaging equipment. I would assume it performs some level of image processing for realtime operations. Nothing complicated is designed by individuals. She may have been the project lead, the medical subject matter expert, or maybe she does also have some engineering background, but that is not stated in the film, to the best of my recollection.

4) She only trained 6 months for this mission - that pretty much guarantees she is not a professional astronaut. She is what in the early 80s was called a "Payload Specialist". The Shuttle would sometimes fly scientists with minimal astronaut training to do experiments onboard the Space Shuttle. Often in Spacelab (a lab module flown in the payload bay). Those payload specialists had little to no operational duties for the Shuttle itself. They were passengers there to do science. They were trained essentially not to break anything.

It is true that people with MDs have performed EVAs for NASA, but those MDs were full astronauts and extremely experienced. Story Musgrave, for example, did actually perform maintenance on Hubble, but it was Story's fifth flight on the Space Shuttle.

There is zero chance that NASA would ever let such a person actually perform an EVA or touch the Hubble. EVA is extremely complicated and takes years of training. They would not risk the lives of the crew or the integrity of the Hubble on such a gamble. So, this is another area where we have to allow for dramatic license of the screenwriters.

This dramatic license also applies to her having flown the Soyuz simulator and crashed it, repeatedly. Space Shuttle astronauts do not receive any Soyuz simulator training. Soyuz simulator training is conducted at GCTC outside of Moscow and NASA astronauts only take that training if they are assigned to fly to the ISS on a Soyuz. Even then, they usually do not get full pilot training. If there are two cosmonauts and one astronaut on the Soyuz, then the astronaut will be in the right seat. That person's job is to operate the radio and life support equipment, not to fly. If there is one cosmonaut and two astronauts, then one of the astronauts will be the flight engineer. They are trained to be a co-pilot, although their primary tasks are vehicle systems. Usually an experienced pilot will get that assignment.