To clarify, does my spouse (the one with the EU passport) need to file this application within the 90-day period?
No. He has an automatic right to be in another EEA country for 90 days and after that he must be a qualified person to be lawfully in that country. If he does that then he has an automatic right to be in the UK under the EEA regs.
As the RC link I gave shows
Who should currently apply
You don’t need to apply for a residence card as a family member of an EEA or Swiss national, but it can:
help you re-enter the country more quickly and easily if you travel abroad
show employers you’re allowed to work in the UK
help prove you qualify for certain benefits and services
You must apply for a residence card if you’re an extended family member.
EEA(QP) Application for a registration certificate as a European Economic Area (EEA) or Swiss national who is in the UK as a qualified personNext, it sounds like our niece needs to file the extended family member RC application. I assume also within the 90-day period.
Yes, as the quote from that RC link above, shows. You can only apply for her RC when your husband can prove that he is exercising his treaty rights. That's why Nan is applying for RCs for herself and her daughter as she has that proof. Have a read of that Home Office link and extended family members links from the RC link I gave, for more information as I'm not sure on the rest.
Lastly, it sounds like my daughter and I don't necessarily have to file the direct family RC application, but according to Nan it's a good idea to do so.
That's in the RC link I gave yesterday and that I have now quoted in this post.
At this moment, as we've been in the UK for 27 days, I assume we are fine with respect to being here legally.
You have a FP with an end date and are here with your EEA spouse.
To understand how the EU Regs work, you and your husband really need to read that Home Office link and the one from gov.uk on RCs. They are much easier to understand than the UK immigration laws that many people on here use. If you don't read them then you will just go around in circles.
Here they are again-
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/588174/EEA-qualified-persons-v4_0EXT.pdfhttps://www.gov.uk/apply-for-a-uk-residence-cardYou can follow the links on this one for an extended family member.
You will also need to keep up to date with the Brexit news and can sign up for texts from the UK government, or read those texts when Nan puts them on here to help others.