If the Battlemage with a Flaming Sword attacks one enemy, and then another enemy dies as a result of the Flaming Sword, does the Battlemage get to draw a card?

Yes. An attack was made, and then a minion was killed. As long as your attacking unit used the Move and Attack ability and then got kill credit, this ability triggers.

Battlemage — Promotional Rainbow

If the Battlemage attacks and kills two enemies in a single fight (if my opponent chose to include multiple defenders), does the Battlemage draw two cards?

Yes.

If the Battlemage activates the 'Move and Attack' ability, doesn't actually attack, and then is intercepted, does that count as 'attacking' the intercepting unit?

No. An interception does not cause an 'attack,' and instead is merely a fight between the unit that used 'Move and Attack' and the unit that intercepted.

If I declare an attack against one target, but then a different enemy defends, does that satisfy this ability?

Yes. An attack was made, and then a minion was killed. As long as your attacking used the and then got , this ability is satisfied.