Does the Stark/Baratheon Brienne stop the Targaryen set dragons from triggering their abilities if she defends against them and loses?

yes, brienne stop any trgger effect when she participate in a challenge, responses are inside the challenge effect.

What about deadly?

Deadly is not a triggered effect, so Brienne doesn't stop it. Triggered effects are any effects that are preceded by boldfaced Response:, Any Phase: or [Name of Phase]:

None of those can be triggered while Brienne is participating in a challenge, no matter if they are on cards in play, cards in hand, shadows, discard or dead pile, or on plots or agendas.

Note that Ambush means "Any Phase: Kneel an amount of influence equal to <this card>'s cost to put it into play" and duplicates give their base card "Response: discard a duplicate to save <this card> from any effect that would make it leave play", so both are triggered effects.