If they have probable cause to believe you are violating the law, they can search you.

Does a will supersede a certificate of inheritance ?

I agree. The will comes before the certificate of insurance. Try talking to an experienced probate attorney, privately.

Possibly, except it was expunged. if for immigration purposes, it will always come up

Is it legally allowed to go on "Leave without Pay" on current job if H4 EAD has expired and waiting to receive renewed H4 EAD?

Depends on your employer. Find out from them what their policies are.

Yes this is possible. Talk to the court about your options and offers.

Do I need to switch from my F-1 STEM OPT card to the I485-based EAD/AP I just received?

not automatically unless you use the EAD/AP, No, you can remain on F1 until you travel.

How do I extended the validity of extended green card as my green card in renewal process ?

Get an I551 stamp. that's your best bet.

How to renew extension on the green card while it's in renewal process ?

Try to get an info pass appointment to get a I-551 stamp in your passport.

What will happen if I was on deferred probation and I got a ticket for speeding again a day before I was done?

Depends on the terms of the order. I agree with the previous answers.

I just had court and I went and plead not guilty for my tickets. My tickets were for no license, no insurance, and tailgaiting?

Try talking to the prosecutor to see fi they can offer you a deal. It is advisable to privately consult with a criminal defense attorney regarding...