In my personal experience, yes you have to "allow" each time you go to different sites in a new browser session, but once you've allowed (in neopets' case, just once each for the site and the mall) it is locked in until you close every last browser window (incognito also acts as a separate session entirely, so you definitely have to allow each time you open it), so it's not as bad as having to do each tab. If you never really close out your Chrome (apart from shutting down your computer) then it's not all that bad.
Explorer on the other hand I can never get to work consistently.
The only things affected by the flash are games, maps, wheels, and of course customization.
Did you ever get the Customization for your pet to work?