NFHS-Appeal after a balk or illegal pitch

Situation 1-1 out and R2, base hit and R2 misses 3rd and scores, batter advances to 2nd

For whatever reason, time is called and and the coach tells the pitcher to get on the rubber and to throw to 3rd when the umpire puts the ball in play. (We know that there can be an verbal appeal while the ball is dead and do not have to put the ball in play)

The ball is in play and the pitcher does not properly disengage from the rubber, balk, immediate dead ball (NFHS) we award R2 third (for the balk). Does the defense lose the right to still appeal the missed base?

Situation 2-1 out and R3, fly ball to RF ball is caught, but R3 leaves early and scores.

The ball is still in play and while having his foot in contact with the rubber, F1 places his pitching on his mouth and distinctly wipes off his pitching hand prior to touching the ball (Illegal pitch-a ball to the batter-NFHS)

Can the defense still appeal after an illegal?