Many people move rooms between tracking and mixing because:
- it can be more expensive to rent a room with a large tracking space if you're not actually going to use it.
- some studios aim themselves more towards tracking - for example, if a room has lots of great mic pres/compressors/eqs, but not many outboard delays/reverbs etc, you might not want to mix there.
- some boards don't have automation - less necessary now maybe than previously, but many people still want it.
- some consoles have total recall, which can be important for some, especially if you're a dedicated mix engineer working for big name clients who expect tweakability.
- some people just like to be in a different space, with different monitoring etc.
As for "which console", people like SSLs for recall, dynamics section, EQ, mixbuss compressor, overall sound, flexability in routing etc. Other people like Neve consoles for the same reason (although modern Neve dynamics sections probably aren't held in the same regard as the SSL dynamics, at least not with the engineers I know).
Some are prepared to sacrifice the flexibility of recall, automation etc for the sound of a vintage console (often used as a big eq-able summing amp).