Are cards in your hand affected by "The Wall" or these in play only?

In other words, can I use "The Gift" to reduce the cost of HB character?

The default in this game is always "in play." You cannot use card effects that are not in play, and effects that are in play cannot affect cards that are not in play. Unless, of course, the effect specifically says otherwise.

Note that "specific" is interpreted a bit loosely to include references to mechanics that only mean something when the card in question is not in play. For example, Army of the Faithful says "Lower the cost to play Army of the Faithful by 1 for each Asshai character you control." One could argue that it does not specifically say that this effect works from hand, but it does refer to a mechanic - playing the card - that can only be accomplished when the card is in your hand, so the effect is active in your hand. Of course, since the "for each Asshai character you control" does not refer to any special cases, only Asshai characters you have in play count toward lowering the cost.

Anyway, since The Wall does not specifically say it affects cards in your hand, it does not. The Baratheon character would not gain the Night's Watch trait until it was in play and thus The Gift will not help you pay for it.