Your mom needs to do an application for you. You have to do it before you are 18 if I remember correctly.
I would stay in country to be on the safe side, but it seems like it might be ok for you to visit or stay with your dad.
I know my husband was traveling for work while waiting for citizenship as an adult (maybe he was even an expat at the time living in Colombia, I can’t remember) but that’s not the same as you living in another country and registering for school there, etc. My niece was in your situation, I don’t know if she visited her father while her citizenship was pending.
I would contact immigration and I would not rely on what just one person tells you there. Maybe you can ask them for the specific information and requirements on their website. This link gives some info, but I didn’t search it to find the exact question you have.
https://www.uscis.gov/family/bring-children-to-live-in-the-US
This is too big of a deal to screw up. I personally would not take any risks. How long is it supposed to take to get citizenship after you apply? I assume it’s less than a year. I think just wait and don’t risk extending the process or losing eligibility. I don’t know how old you are.
Just my opinion. I’m not a lawyer or immigration expert.