Can I get legal resident status after living as illegal immigrant?

It is possible to obtain permanent residents status after falling into unlawful status in the US. Consult with an attorney to discuss your options.

I am belong F4 category. My sister sponsor me and priority date is 2-Jan-2003. Can I expedite my F4 Visa Interview.

You cannot expedite the interview until a visa is available for your priority date.

Does an LPR have to apply for reentry permit if going abroad for 10 months care for sick relative?

It would be a good idea to obtain a reentry permit under these circumstances.

Travelling to India while OPT STEM Extension is pending ( travel dates: 05/16/2014 -06/14/2014 ) OPT EAD expires: 06/02/2014

Travel should be the lowest of your priorities at this stage.

I have lost EAD card in mail, then i reapply for replacement of it, they denied that my EAD is already approved . what to do ? i

You need to contact USCIS and request a replacement for the lost card.

I am in process of filing my citizenship. Is it advisable to change the employer during the course of this process?

Changing employers should have no bearing on your naturalization application.

Am I eligible to file for naturalization after 3 years? I had divorced the abuser US citizen husband.

Yes, you appear to qualify for naturalization under the 3-year rule.

Is it possible to change my status based on the following

This depends on whether you have been maintaining your F-1 status in the US.

Today i receive a msg from uscis

It should mean that an approved visa petition has been forwarded to a US Consulate overseas for visa processing.

You can use the married name and correct the SS and other documentation when you get the work permit.