QIf I play Andromeda's Fanfare ability, then put a Humpty Dumpty onto my field and evolve it, are both my Andromeda and Humpty Dumpty immune to the damage dealt by Humpty Dumpty's On Evolve ability?

ANo, only your Andromeda is immune. Andromeda's Fanfare ability gives "This doesn't take ability damage" to each follower on your field at the moment the ability is played, which doesn't include your Humpty Dumpty. If you put Humpty Dumpty onto your field before Andromeda, then evolve Humpty Dumpty, both will be immune to the ability damage.

QIf a follower on my field has "This doesn't take ability damage," and an Agnes Tachyon gives it -1/-1, does it still lose attack and defense?

AYes. Being given -1/-1 is different to taking ability damage.

QWhat is ability damage?

AAbility damage is all damage except for damage dealt by two followers in combat or by a follower attacking a leader. For example, damage dealt by a spell is ability damage.