I'm confused as to why the EAD application was submitted 1 week before the H-4 extension application. With COVID, there is no 'normal'...

Talk to the school's "RO" ... if there is no restriction, you should be able to change to H in the US, or by going home for stamping.

Will I be able to start graduate program if I-539 for H-1B to F-1 COS is pending?

I agree with Mr. Shusterman, don't be lazy and don't try to claim that COVID-19 makes it impossible to go home for the visa stamping. Otherwise,...

Yes, once you get the actual I-130 approval, the I-485 "kit" can be filed.

Can dependent (on mother) kids study in any state if the parents work and live in different states

Yes, they can study in TX on L-2, or file a change of status to H-4.

Can new employer can file for H1 COS while my existing L2 extension is in progress with USCIS & waiting for results?

Q: can file for change of status from L2 to H1 to work for them directly? A: You didn't mention if you previously had an H-1B. Q: This H1...

Are you saying that she left the US shortly after getting her greencard? If so, her situation may be at risk. Did she apply for a re-entry...

How do I renew my work authorization with my i485 administratively closed?

Hire a lawyer ... maybe he/she can get the I-485 transferred to the immigration court.

How can I get a work permit?

As Mr. Behar stated, there are many options. He should have a Skype consultation with a US-based immigration attorney.

Will a H1-Transfer applied after 60 day unemployment grace period gets approved?

No, if the CIS officer reviews the papers closely, it should not be approved for you, but can be approved for the employer. Then, as Mr. Behar...