Can Brogan use his red card just to melee attack somebody adjacent or does he have to move 2?
Page 5. Point 1:
"Some cards have mandatory movement as part of the card's text.
This is not considered to be regular movement and you must move as far as the card dictates.
You may not resolve an action, unless you can perform this mandatory movement."
Also on his ongoing blue effect (protect minions from basic attacks) does the effect take place immediately when he reveals the card or from when initiative reaches him to actually play it?
You can only resolve an action when it's your Initiative. Until then, nobody knows which action you want to resolve. Chances are you will use that card to Move, instead of using the Skill.
Don't forget that if you chose to resolve a Skill (Shield in this case), you won't be able to use movement action from the same card.