ILR Long residence query about Kid born in UK

Hi guys
I am confused about the ILR process for kids and need your advice please.

I have been in UK from Nov-2010, so planning to apply for ILR through Long residence (No issues here)
My wife is with me since Nov-2010, but has a break of more than 6 months twice in the last 10 years, So I am planning to apply for FLR. (This is fine too)

I have 2 kids and my questions are about my kids application.
1) First kid born in India . So need to apply for FLR for this kid along with my wife, correct?
2) Second kid born in UK. Can I apply for British Citizenship for the kid once my ILR is approved. If yes, then another question. All of our current visas are ending in Nov-2020. So if my ILR application takes few months , then my kid will be without a valid visa until that time. Is that alright?

Please advise.

1) First kid born in India . So need to apply for FLR for this kid along with my wife, correct?

Correct yes. Your wife needs to meet the English requirment and you need to meet the £22,400pa financial requirement.

2) Second kid born in UK. Can I apply for British Citizenship for the kid once my ILR is approved. If yes, then another question. All of our current visas are ending in Nov-2020. So if my ILR application takes few months , then my kid will be without a valid visa until that time. Is that alright?

2a. Correct.
2b. A UK born child has an entitlement to register as British once either parent has ilr. They do not need a valid visa or ilr for the application.