Children Eligible? What steps/documents need?

Hello! I'm seeking advice as the laws seem to have changed recently and its all quite confusing!

I am German by naturalization. My paternal grandparents were German, so my father, siblings, and I (all born outside of Germany) all applied for naturalization and received naturalization and passports ~15 years ago.

My wife is not German and we do not share a last name. Out children were born outside of Germany. They have MY last name.

I believe they are eligible for citizenship, but am unsure of the next steps...

Here are my questions:

1. Do I need to do a name declaration and/or birth registration first before I can apply for a their passports? Does the name and the fact that my wife is not german matter?

2. In addition to all of my children's birth certificates, which of MY documents will we need? Naturalization certificate? Passport? Birth certificate? Also my wife's non german-passport and/or birth certificate.

3. Will I need any of my father's documents?

4. Any advice on getting a consular appointment? The New York Consulate covers my region but it seems very difficult to get appointments!

Danke!