1) Does my wife REALLY need a national registration number separate from my national insurance number for her EEA Family Permit?
No, she doesn't (it even says so on the UKBA website, on this page:
http://www.ukba.homeoffice.gov.uk/eucitizens/documents-eea-nationals/).
It also says so on the EEA Family permit paper application form (
http://www.ukba.homeoffice.gov.uk/sitecontent/applicationforms/eea/eea21.pdf):
Note 1. There is no requirement for your EEA national family member to apply for a registration certificate and we can decide your case without them applying. However, we must receive evidence of theiridentity, e.g. passport or identity card. If your EEA national family member wishes to apply for a registration certificate, they may apply separately using form EEA1.
For her EEA family permit application, all you need to do to show that you will be exercising treaty rights in the UK is provide your EEA passport (or a certified copy of it) and evidence of your job in the UK (job contract, letter from employer, payslips if you are already working in the UK etc.). Plus, you need to provide your marriage certificate to show she is your non-EEA family member.
2) If I go in person in Croydon will I be in and out quickly and will I be able to keep my passport?
3) If I mail in the application with my passport how long will it take to return it?
You don't need to apply for it at all, so this isn't an issue.
4) If I don't have my passport back in time to fly to the US and I leave and return on my US passport and not my EU passport, will I cause an issue? I suspect yes, since I'm here on the EU passport. Would it have been enough to have entered the country on the EU passport once already?
It's a moot point, because you don't need to apply for EEA1, so there's no reason for you not to have the EEA passport, but for your information:
You would have no choice but to use your US passport to enter the US (as it is illegal not to, if you are a US citizen). For the UK, it's not mandatory to use the EU passport, but if you did you would need to prove some other way that you were an EEA citizen and legally allowed to live in the UK.