The bonus that the target creature receives is locked in as Fists of Flame resolves, counting the card drawn during its first effect. The creature won't get an additional +1/+0 if you draw another card later in the turn.

If the target creature is an illegal target by the time Fists of Flame tries to resolve, the spell doesn't resolve. You won't draw a card.

If a spell or ability causes you to put cards into your hand without specifically using the word "draw," Fists of Flame's effect doesn't count them.