Effectively the question becomes: Can you add weapon bonuses and benefits from supernatural abilities to spells that act as weapons?
The flame blade spells says it is wielded "as a scimitar". So I would assume all feats and abilities that function on a scimitar will function for flame blade. So yes both judgement and bane effects can affect a scimitar so it can affect a flame blade that you wield.
I don't see why not. Although if the spell duration expires or is dispelled, that weapon is gone.
Doesn't matter so much for Judgement, but Bane would not automatically apply to the next Flame Blade spell you cast,
as that is a different weapon, just like if you drew another scimitar after one is disarmed.