Right. Her legal resident status will not hinder her in gaining custody, unless you have reason to suspect or she indicates that she may be taking the children to her native country.
1 & 2 refer to serving an Answer to the Complaint. Your signing an Acceptance of Service is not the same thing as an Answer. A of S just shows that you were in fact served with the Complaint. You do not have to file an Answer if you don’t want to. However, failure to file an Answer means the court will deem you to have admitted to all of the claims she made in her Complaint. If there is anything you disagree with, you should file an Answer denying those allegations. Additonally, you should consider whether or not you need to file counterclaims within your Answer. If you need more help with this process, the Rosen Online service (rosen.com/DIY) may be a good resource for you.