Getting Family Permit

Good morning all,
My husband and I are in UK for 10months and we were trying to get him Residence Card but unfortunately his application was refused, and now that his visa is expired, he has to go back to US. I am Polish and living in UK for 6 years. He's American.
He is leaving now and he will apply for 6 months Family Permit to come back to me, live and work in UK and then make another application for Residence Card.
Please let me know what documents does he need to take with him and what actions he has to take in USA to get Family Permit and come back to me with work permit? I am 5 months pregnant - is this going to help him to come back to UK?
Also - do you know how long does it take? His application was refused 3 times already because we didn't send enough\correct documents. He will stay in US for 2-3 months anyway so he can gain money for his return ticket. Is it guaranteed that he will receive Family Permit and visa in this time and he will be able to return to UK?

It's a really simple application - since he already has the automatic right to live in the UK because you are living and working in the UK, the permit is just a formality.You being pregnant has no bearing on it whatsoever (though you can mention it).

All you need to prove is that
a) you're married
and
b) you're an EU citizen working in the UK

So, you need to provide:
- his passport
- a copy of your passport photo page (preferably certified, but I think a regular photocopy may be okay)
- your marriage certificate
- evidence of your job to show your are exercising EEA treaty rights in the UK (job contract, letter from your employer confirming your employment, payslips and bank statements)

You should also include your refusal paperwork and a letter explaining what happened.

See here: https://www.gov.uk/family-permit/documents-you-must-provide

Generally, an EEA Family Permit takes about 2-3 weeks to be processed (though it can take anywhere from 1-2 days to 6 weeks), though you can purchase the priority processing service for $228 to have it put at the front of the line (http://www.vfsglobal.co.uk/usa/user_pay_services.html). With his previous refusals in the UK, they may need some extra time to look into the reasons for those refusals, so it could add 1-2 weeks onto the processing time, even if you use priority.

Thank you very much for your reply ksnad24!