Problem #2: if the inactive tab is paused and the game is unable to run normally (it's forced to pause or to skip steps) it may result in a corrupt result which - in case of a duel with cash at stake - might seriously affect the contest holder's reputation
Problem #3: I am unable to replicate the Internet Explorer popup and I don't remember what was it about, but it seems once you choose something it will force a refresh and won't pop up again.
As for Problem #1, it may be a difference between browsers, but that doesn't make the final result any better. One choice would be to make the games unavailable for specific browsers, but from experience I can say people would rather ditch the game than play in a browser they're not comfortable with.
Construct is a really awesome piece of software and it's taking me 3x longer to do the same things in Flash compared to C2, but for now this seems to be the better choice. Large skill gaming websites, such as king.com, are still using Flash for their new games, and they probably do it for a good reason.
The time for HTML5 will come, but not just yet.
I'll have to put my name on the list of people who dream of a swf exporter with C2 :)