When all of your characters are being ignored you are being ignored you have initiative until the point at which you are not being ignored. So yes you can play cards and you retain initiative until something changes (banding a non ignored character into battle such that the initiative would shift or playing a card that interrupts or negates the ignore enhancement)
Secondly you can target any character that is not ignoring your character (or otherwise protected from your character) so you could devastate another player's territory, even the ignoring player (assuming you could normally)
Let me give you a quick example:
Player A attacks with King Solomon (9/11) player B blocks with a Black Character, it really doesn't matter who but for clarity, let's say the new Ishbibenob (10/10, ability irrelevant). King Solomon plays Royal Parade (Interrupt the Battle and Ignore one Evil Character) this makes Solomon ignore Ishbibenob, who now get's initiative (since he is losing by ignore). Player B notices his Wrath of Satan (Discard all heroes) card in his hand and decides to play it. This discards all heroes play, Solomon survives because he is ignoring Ishbibenob and perhaps some other characters survive due to protection. Player C can only watch helplessly as his unprotected heroes are discarded.
Since Ishbibenob is still losing he has initiative to play Babel to bring in a few other evil characters (who are not being ignored) such as the Saph he has in territory (9/11 Black character who prevents the next enhancement opponent plays), this would pass initiative, or he could play Devourer to interrupt and discard all good enhancements in battle (effectively negating the Ignore) this would also pass initiative.
I hope this makes sense. Although to answer your questions, yes, you can devastate your other opponent's territories (even the one doing the ignoring) and you will not pass initiative unless you either bring more characters into battle (that aren't being ignored) or interrupt or negate the ignore or become Cannot be Ignored I suppose.
