batting out of order?
 

An improper batter (someone batting out of order) completes at bat and gets out. Defensive team notices than appeals. Is the batter that failed to bat in the proper order out too?

Thanks

If dicovered before the next pitch (legal or illegal), the out would be recorded to the proper batter/runner and the batter immediately following the proper batter would be required to come to the plate for her turn.

So, the out would not be credited to the improper batter who already got out. So, only one out would come of this even though the improper batter already got out and the appeal was made before the next pitch. The out would be credited to the proper batter who failed to bat in his/her spot?

Correct. The proper batter is called out, and the improper batter removed from the bases. The next batter is the one that follows the called-out batter's position in the batting order.

Yes
 

Yes. The correct batter (CB) would say to the improper batter (IB) as she returns to the dug out, "Thanks for hitting for me. I wish I would have been safe."

Then the next batter who follows the CB in the line up would swing away.

Even if it is the improper batter

Doesn't the out stand in ASA?

Yes. Its entirely possible to remove the improper batter from a base and put him/her at bat to try again.

The improper batter made an out. Do you have an out for the play and an out for the BOO is the question.

FED rule 7.1.2 effect: The umpire shall declare the batter who should have batted out (not the improper batter). Any other outs on the play stand.

except that an out for BOO supersedes an out made by the batter-runner. All other outs gained on the play stand.

Not sure what you are saying here...could you reference a rule either FED or ASA?

That's the point I was making

My emphasis:

FED rule 7.1.2 EFFECT: The umpire shall declare the batter who should have batted out (not the improper batter). Any other outs on the play stand. All runners who were not declared out must return to the base occupied at the time of the pitch....

Short answer to original question:

In ASA - yes
In NFHS - no

Not sure about the rest of the alphabet soup (NSA, USSSA, AFA, USFA, etc...)