I'm having the same issues with being unable to start a quest in quest mode. After I choose party and difficulty it hangs on the loading screen.
One note to make, I went ahead and uninstalled/reinstalled the game. When I loaded back up (which nuked all my local characters/parties/progress), I went immediately into quest mode (with no progress in Story Mode), selected a party of new adventurers and it let me into a quest. Which I promptly failed because it only had 8 turns on the counter to make progress in 5 locations. After that I tried for a second quest, got the hung loading screen again. I reboot the phone, got in again, and got in to another quest (again with all of 8 turns), lost, tried to load a third quest, got stuck on loading screen and haven't been able to get back into a quest after four reboots.
Before I uninstalled I tried quest mode five times and got stuck at loading each time. After reinstall I can seem to get in, but I have to reboot the game every time and going through the quest pops up all the newbie tool tips each time. I have not tested what happens if I get some experienced characters in story mode and then try again to load up quest mode to see if it hangs like it was doing before the reinstall. It feels an awful lot like there's some bad interaction going on between versions of characters for quest mode and story mode causing hangup issues.