Does the Temporary Protective Status affect the Asylem case in any bad way?

You are permitted to file for both asylum and TPS.

"What is the waiting time"

The waiting time is workload specific but in about 3-5 weeks after you provide them with the medical and report.

If amended Petition gets rejected, then will the older H1-B Petition still be valid? I have one petition only, not any VISA yet.

Yes your old petition is still valid and may be used.

Do i need file my passport application first or can i simultaneously file mine and my 3 children's?

Yes you can but it is highly recommended that you file an N-600 as proof of citizenship is part of everyday life.

Selected as the best answer

Contact NVC directly and they can make the change.

Working on a laptop while on vacation is ok but short of that, no. You may not make personal business contact with anyone.

Is there anyway to speed up my I-589 application process ?

There are no ways to expedite an asylum application. Your timeline is consistent with the norm and actually may take longer given the country...

Not sure when to file for removing condition on green card. Does being have anything to do with that?

You can file 90 days before the card expires. You can begin reviewing what you will need to file now.

Can a 2 years conditional green card holder be a co-sponsor for an AOS application?

That is no problem assuming he meets the financial requirements.

I-130 through sister Approved but priority date could not be established

How do you know the I130 was approved in 2009?