I slightly disagree with first replies:
1/ Yes. If one TL is set at TOGA, A/THR will not be active anymore (A/THR blue), one ENG will provide TOGA thrust while the other will be at max CLIMB thrust.
2/ What is engaged? A/THR is either armed, active or off. In your example, it will go from active to armed mode.
(Eric, A/THR can be active in thrust mode...cf. all (most) climb are flown with thrust mode and A/THR active)
3/ No, cause A/THR is no longer active.