Admit Until date is your I-94 expiration date, right corner date is FORM expiration date. you can print I-94 online from CBP website

I am a U.S Citizen & would like to petition my two children ages 17 and 20 who have been living w/ me after their visas expired

Yes, you can file Form I-130 and I-485 concurrently, they will be able to adjust status in USA. must be filed before kids turn 21 years old.

I 130

I-130 does not need your mother's address unless you are filing for her.

Left employer on bad terms. Can they cause problems on my future h1b transfer ?

If you have not violated any terms of your H-1B, it does not matter. Exactly what will they report.

Can an LPR petition her children (16 and 19 years-old) while they are here in USA as tourists?

Yes, you can file I-130 for your kids, but the visa availability dates are not current right now. You can not file I-485 until visa becomes...

Is there a separate category for a permanent residency if I am offered an employment by a local city government?

There is no different way of doing your residency or any expedited path except Eb-1 category if you qualify for that.

If your company is filing extension of L-1B visa and does it before expiration of I-94, you don't have to leave USA and L-1B extension has nothing...

Yes, you can do it so far as you don't get paid for it and maintain your status

I-130 can not be expedited and there is a long wait for visa availability

Currently I am on F1 OPT and want to apply for Change of Status (COS) to H4 as well as for H4 EAD. Can I do it simultaneously?

As per USCIS guidance, you can file it concurrently but make sure you send USCIS fee for both forms.