The deckbuilding restriction refers to your draw deck. Plots with the House Arryn trait (such as Wardens of the East) will not count towards the 12 House Arryn cards you must include in your deck, and neither will Banner of the Falcon itself.
Cards that are not in play do not count towards the number of House Arryn cards you control. That includes Banner of the Falcon, since agenda cards are not considered to be in play. Your revealed plot card is considered to be in play though, so it will increase your number if it has the House Arryn trait.
A quantity cannot be reduced so that it functions with a value below zero. If your initiative value is a negative number, it will be treated as 0, so your loyal characters will not get +1 STR if you control no House Arryn cards.