To expedite the process you can submit a new corrected application. However, it's unlikely that you would receive the fees back for the first one.

Many different factors come into play to properly answer this question. When did you enter the US? Also, keep in mind that having US children under...

Yes, you may stay in the US and continue working with the approved EAD for your pending I-485 despite your visa expiring

Yes, you can apply for F1 even with your H4 in administrative processing

You will be able to file after spending a year out of the US

The university should apply for you. Yes, premium processing is available.

Yes, your wife would be out of status if you become a green card holder if she's currently on H4

Your aunt should be ok as long as she departed before the denial. She should bring evidence of her return home.

No, you should not have an issue filing for a GC after your L2 is filed

The government has no responsibility nor incentive to support you in this endeavor. I recommend you research potential scholarship or financial aid...