Names "issues" for kids getting US passports

[Mamed 18]

Trying to help family friends here, and need you advice please. 

So here's the situation. The mother recently became a US citizen, (father will have interview by the end of this month), and she wants to get US passports for her kids too (She didn't apply for herself yet. Wants to do all at once, is it possible?).

Now, here are the name issues:

1. the oldest son has his moms last name on his birth certificate, but his father's last name on green card (before family immigrated (refugees) his father put his last name for him on some papers (just like that) so, now on his green card (and everywhere else in the US) he has his father's last name.
2. the second son's first name is misspelled by the immigration crew (O instead of U), and now on green card (and everywhere else in the US) he carries misspelled first name (birth certificate is fine)

 

now the questions are:

1. for the 1st kid, do parents have to do anything if his last name is different on two documents if they want him to have his father's last name (the one on green card)?
2. for the 2nd kid, how to fix his first name for the passport if it's misspelled on green card?

 

Thanks in advance

Edited March 3, 2018 by Mamed