Can my work authorization be renewed after I file an appeal

Technically, your case is due to be adjudicated and hence you can apply for EAD..

No, i wish there was such away to pay additional fee for a faster processing. Unlike, other visa's such as L's and H1b's you have to just wait.

I have very complicated F-1 student status trouble, and I am currently "Out-of-status" need some advices for it.

Yes you have several issues here and i suggest that you speak to an experience immigration lawyer near you. Since you are out of status for only...

Can i adjust my status again or what is the best way to fill the R visa with sponsor of that organization

You are using a wrong terminology. Yes you may change status from F-1 to R-1 but consult with immigration attorney before change status.

I am not sure what is you asking here.. could you elaborate please.

Medi-Cal (Medicaid), New Immigrants

Clearly, she would have to be at certain age and US citizen. Depends on each state policy i suggest that you contact your state medical office and...

Is there an additional cost for this; or is it inclusive whenever you file I-485 application?

Every Alien filing for Immigrant petition must be finger printed except children under 14 years of age of adults over age 75.

Removal proceedings - immigration

You do not seem to be in removal proceedings. You may ask for a copy of your file through a process called FOIA. To know more go to USCIS.gov/foia

My wife filed for asylum timely in USA her case got denied all motions and appeals were timely filed and finally she got denied

Your wife is subject to 10 year bar. However, now that you are a LPR, a waiver of extreme hardship can be filed on her behalf. We do lots of...

Immigration

No, no 240 days extension..you need to consult with a business immigration attorney ASAP..