I also noticed that quest progression stops sometimes when idling, I suspect some messages are the cause. That makes teh game essentially pointless to idle as you can leave it for hours and the resource you need the most (Leadership) has not advanced.
Quest progression should continue as long as the game is open, AND in an active tab. Browsers (Chrome and I think Firefox) won't have flash run while you are on other tabs. A workaround is to open up the game in a separate window. Lmk if the issue continues otherwise. Random events shouldn't interfere with quest progression at all, if that's what you mean by messages.
 "Karting Racing"){kind=icon}
 "Tip of the Tongue"){kind=icon}
 "The Swemp Mnster"){kind=icon}
 "Stacko Level Pack"){kind=icon}
 "Guanarito Virus"){kind=icon}
 "Find and Clean"){kind=icon}
 "You Find Yourself In A Room"){kind=icon}
 "Rullo"){kind=icon}
 "Zenge: Starborn"){kind=icon}
 "What's inside the box?"){kind=icon}
 "Strange Magic Escape"){kind=icon}
 "Aviator"){kind=icon}
 "SGARDEN"){kind=icon}
 "Other Worlds"){kind=icon}
 "The Royal Archers"){kind=icon}
 "Reach the core"){kind=icon}
 "Autumn Mansion Escape"){kind=icon}
Quest progression should continue as long as the game is open, AND in an active tab. Browsers (Chrome and I think Firefox) won't have flash run while you are on other tabs. A workaround is to open up the game in a separate window. Lmk if the issue continues otherwise. Random events shouldn't interfere with quest progression at all, if that's what you mean by messages.