No doubt somebody with more specific information will be along to answer your questions. However I think the B767 you alude to, should really be a B757. Certainly that is the equipment type shown on the airports website. I have seen 767's at INN but they were limited load flights many years ago, and more recently a 767 operated privately by a Russian oligarch. There are limitations imposed by the short high elevation runway, and turn radius constraints at Innsbruck that do not suit it to high density passenger 767 operations.
If a flight had a flap problem, you can be assured that Innsbruck wouldn't be the first airport you would choose to land at. However if the problem was resolved then that might no longer be an issue. Weather at Innsbruck although statistically good, can be a very real problem on occaisions. The terrain and approach / missed approach procedures impose increased restrictions on flights to this airport. The difficulties associated with Innsbruck classify it as a category "C" airport which imposes additional requirements on airlines that operate there. This normally involves additional annual simulator training specifically tailored for this airport. Crews operating there are also normally well experienced and selected for this route.
I am really not sure what you mean by the sentence:
It was said that the Captain told the passengers that the flying time was 15-minutes and then informed them that they had 15-minutes fuel which meant there would be no further delays.
All aircraft are required to divert when their reserve fuel levels fall to a level that amounts to the fuel required to complete the diversion
plus an additional 30 minutes as well. Perhaps the Captain had communicated that he could only continue holding for another 15 minutes before he would need to divert to Munich and the flight time there was 15 minutes. That would certainly make a great deal more sense. However I wasn't there, and by the sound of it neither were you or your fellow passengers, if indeed you were all sat in the Innsbruck departure lounge, waiting for your flight home? I'm not sure therefore how this third hand information arrived into your seemingly excited group?
Beyond the erroneous and the imaginative, I cannot see what the problem was other than the frustration of delays brought about by a poor weather day at Innsbruck airport? The upside of operating to INN is that it is much more convenient for many of the ski resorts it serves. The downside is that in poor or marginal weather it quickly becomes subject to disruptions that can cause lengthy delays and consequential frustration.