Is she able to leave the country with our child?

She can't legally get a US passport for the child w/o your signature. Get a family lawyer and sue for joint custody ... IMMEDIATELY.

Can I work with EAD when I am on H1B with the same employer?

You forgot to tell us how you qualified for an EAD.

Pending asylum with work permit waiting for interview rescheduling, can I look for a one-year or more engineering job?

Yes, you can look for long-term work. USE A LAWYER if you want to increase your chances of getting an approval.

Do I have to notify USCIS of changes of my reduced work hours?

Yes, the employer, not you, must file an amendment, using for I-129H ... other wise they must continue to give you full pay.

Here's the problem. Your employer can not cut your hours without filing an amended H-1b. As for unpaid leave, you can file to change to...

With the COVID-19 problem ... I'd say that it will take a minimum of 6-9 more months.

How do I file for my wife who is living in the USA after her visa expire

If you are a US citizen you 2 should file I-130/I-130A, I-485, I-765 & I-864

7th year H-1b extension

Yes, your employer should be able to file for a 3 year extension ... talk to their attorney.

H4ead Premium available?

Sorry, H-4 has never qualified for premium. And, H-1B premium is temporarily on hold.

Do you have 2 passports? I suggest the Consulate closest to your last place of foreign residence.