I tried ⌘tab / alttab / esc (that only opens the menu). It's hard to check the internet (friends/work/emails/etc) while in full screen and needing to save and exit every time. Running on OSX ...
After the Twins' Mansion, the Cave Tour continues to a set of stalactites/stalagmites. With the time traveler, apparently it is supposed to be possible to use her ability to phase forward through them....
I already played the cave two times, so anly the scientist is left. unfortunately you have to pass all constant caverns to reach the Scientist's Silo: So in my third game I will take the monk and ...
On the diagram below taken from here the orange areas are character specific. Can you skip them, even if you have the appropriate character on the team?