Batting out of turn?
 

With two outs and a runner on first, a batter steps to the plate out of turn. While he is batting, the runner is picked off for the third out. No appeal is made.

Who bats first in the next inning?

Whoever wants to. A team may bat out of order all day long, as long as the other team does not appeal. Sounds far fetched, correct?

The correct batter, bats first in the next inning.

If the team had appealed during the incorrect batters turn at bat, the correct batter would just replace him at bat and assumed his at bat with the count that exists.

The proper batter should be brought up, but if the improper batter bats he's certainly subject to appeal.

Had the ball been batted into play and the 3rd out recorded, the defensive team has until the next pitch of the next half inning to appeal and have the proper batter declared out. Once the next pitch is delivered, the improper batter's is legalized, and we go to the next spot. If they appeal, proper batter is called out, and we resume the order underneath him.

Since the improper batter did not finish batting when the inning had ended, and if the defense appeals, the proper batter can't be declared out, could he? The runner is still out, and the original proper batter should bat in the next inning (I assume).

But what if the same, (or another) improper batter tries to bat again in the following inning? Is the appeal from the previous inning still in affect?

The improper batter did not complete his time at bat and he did not become legalized. He is not out on appeal. If the improper batter led off the next inning, he would again become subject to a BOO appeal, and there would be no penalty if the appeal were made before he completed his time at bat (proper batter would assume the count).