What is the maximum apogee a Space Shuttle (and which one of the five space-flying Shuttles?) ever reached and what is the maximum one it could have reached? A Space Shuttle never exceeded the apogee record of Gemini 11 (1,368 km / 788 mi) which, till today, hasn't been broken yet. Did a Shuttle ever achieve another orbit than LEOs and parking orbits?
The highest orbit the Space Shuttle was designed to achieve was 643 km. Theoretically one could do a more elliptical orbit, 200 km by 1000 km or so, although I suspect there would be some thermal problems, and it might be too difficult to reenter. This comes from taking the designed max and mostly splitting the difference between apogee and perigee. In any case, this would have been well below the Gemini 11 mission, which orbited circular at about 1370 km.
The highest actual Shuttle mission was most likely STS-31, which initially deployed the Hubble Space Telescope, at 631 km.
Partial answer (to
What is the maximum apogee a Space Shuttle (and which one of the five space-flying Shuttles?) ever reached )
The maximum altitude reached by a shuttle orbiter on an unclassified1 mission was on STS-82 (Discovery) after the 3rd reboost of Hubble. This resulted in a 335.1 X 321.0 NM orbit. (620.6 x 594 km)
1 Published orbits for the classified shuttle missions are lower than this.
Reference: Space Shuttle Missions Summary