Specifically say I have The Wall (TWoW) out would Baratheon characters in my hand be considered to have the Night's Watch trait and be able to be reduced by cards like The Gift (EB).

Similarly if I have out one/all of the Night's Watch Agendas could the Street of Steel/Sisters/Shadowblack Lane locations reduce a Night's Watch character's cost?

No, they don't. Your hand is an out-of-play area and the cards in it don't interact with in-play cards (and viceversa) unless they say so.

Thats the direction I was leaning in but wanted to be sure before I broke the news to my friend and his new Night's Watch deck, thanks