The H-4 should have been filed with the H-1B extension. Be careful he is not acquiring unlawful presence by filing late.

In theory, yes. But there are COVID-related restrictions. See https://mx.usembassy.gov/travel-restrictions-fact-sheet/

You can work with your valid OPT authorization until expiration. You can also work if you apply for an EAD assuming you are eligible to extend...

Conviction of molestation of a child could get him deported. You can also report him to USCIS for immigration fraud. USCIS has a tip line for that...

The case being dismissed does not expunge it from your record for citizenship purposes. Seek legal advice before proceeding.

That is a question for the HR department. Military clearance may be needed which has nothing to do with your having an EAD.

It is an individual taxpayer identification number assigned to individuals here legally who are not seeking to work. You can apply for it online...

You cannot control when USCIS will reschedule. Usually a month later. Staying away for under a year is not the law for maintaining permanent...

An E-3 is intended for a professional requiring a Bachelor's Degree. This is problematic for a nanny position. A J visa is still the normal course...

This will not affect your green card process. Should your home address change, please use Form AR-11.