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HOH filing status is only an option for a married person if they live separately from their spouse, like if their spouse lives in a different country. If your friend applied for N-400 under the 3-year rule, filing taxes as HOH would be fatal to the application as it implies that they do not meet the requirement for continuous marital union.
The case will move on to NVC where it will sit for a decade, possibly two. No action required from you until PD is current on Table B of the visa bulletin.
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Technically, your wife is out of status, now that her I-94 has expired. But with a pending I-485, she is authorized to stay, even if she is out of status. Just make sure she carries a copy of her I-485 receipt notice whenever she leaves the house.
Actual case on why filing after 3yrs+1day reduces risk of N-400 denial -- https://www.visajourney.com/forums/topic/781869-n400-case-denied-for-3-years-marital-union-after-evidence-submitted/?do=findComment&comment=10656014
I-130 processing is just the 1st stage of the process. After that comes NVC review, then the wait for an interview at the consulate. Whether you keep to K1 or start over with CR1, it's not going to be fast.
Both the petitioner and the beneficiary need to submit a signed statement of intent to marry within 90 days of US entry. Did you both submit one with the I-129F? With the right magic words?
In that case, his SSN card should arrive in the mail within 2 weeks of his US entry, assuming the DS-260 has the correct US address. If the card is not delivered after a month or so, book an appointment at the local SSA office to apply for SSN in person.
My interview was in 2021. USEM has not posted a more recent update on their website, but as of May 2022, they noted that DQ to IL wait time for immediate relative categories is around 2 months -- https://ph.usembassy.gov/visa-update-consular-operations-visas-as-of-june-2022/
You mentioned your mom officially surrendered her GC. Did she not already submit an I-407 at that time? If she already submitted it then, just attach a photocopy of that I-407, so USCIS doesn't get confused about your mom still being LPR.