The children would have to reside in Arkansas for at least six months before mom would be able to file any custody/visitation action in Arkansas, and possibly not then if the father can show the Arkansas court that there is cause to decline jurisdiction.
If by "surrendering custody" you mean voluntarily terminating the parental rights, doing so would be difficult at best in any state unless it is relation to an adoption proceeding.
With that said, in any state you can have a suit in which one parent seeks custody over the other and the Court award custody to just one parent, in which case the court may or may not award visitation to the other parent.
As to prosecution for adultery, if you are referring to a criminal prosecution instead of grounds for divorce, check with a Colorado attorney. Although I am not familiar with Colorado law on this, many states no longer consider adultery to be a crime.
I presume Amy's threat of prosecution for adultery is based on potential trouble he could get in with the military. As Scott already pointed out, it is hard to believe Colorado would still have criminal prosecutions for adultery. Amy could be awarded custody in Arkansas, but the home state for determining custody is the state where kids resided last 6 months.
Arkansas courts typically provide the noncustodial parent visitation rights with the children except in cases of abuse. The amount of visitation may change depending on the age of the kids. I think Aaron needs to consult an attorney sooner rather than later to protect his rights with regard to his children.