Can she do that?

The fact that you have two small children doesn't change that you are probably on a month to month tenancy where 30 days notice is sufficient.

What can i do to stay till June 17th ? and if i applied for school will it effect my H1B application

Just bear in mind that reinstatement usually needs to be completed within 5 months.

Am I eligible for cap exemption for 2016.

Yes you are still eligible according to your statements.

Can I travel outside the USA while on I'm on OPT?

You need a valid visa in order to enter the United States.

I would just like to add that there are many legal aid services organizations that provide this assistance free of charge - do a quick google...

How Can I change my status from F-1 visa?

From your statements the best way would be to find an employer to sponsor you for an H-1B. Good luck!

Will it be possible for my wife to use the approved H1 petition to return to work from H4 without getting into cap and lottery?

Attorney Behar has given excellent advice and you are urged to take it.

If he apply for me with this form, should I be illegible to be here? or I should accept with form 797? Or after form484?

Your brother can file but the waiting line for siblings of U.S. citizens is several years long.

H1B visa process timeline?

The first day USCIS can receive the petition is April 1, 2016.

How long it usually takes to get an answer in the mail? ( Officer told me that up to 120 days - but I hope it is not realistic)

It is realistic, try to be patient and keep your fingers crossed for a quick approval.