Yes you may sign on behalf of your minor child.

If it is since corrected it shouldn’t be an issue.

That the filing fees were waived and the case will begin processing.

Yes obtaining a medical from another USCIS civil surgeon is not an issue.

You may notify the officer at the time of interview without issue.

Consult with an attorney to determine if CSPA applies.

Yes you may and will need to disclose the trip prior to oath ceremony.

More information is needed about your manner of entry.

Attend the appointment and contact USCIS thereafter.

This does not appear to be a refusal for DS260 purposes.