Is there a way to challenge an illegal reentry charge after being deported by CBP and not an immigration judge?

You should have your immigration attorney assist as they are in the best position to help you.

Papers

This is not an immigration question. Should be answered by a landlord lawyer.

How much time staying out of US is counted on LPR ?

You should be okay if you only leave for 3 months. It is important to realize that there are many more requirements than physical presence before...

Processing times can vary depending on background checks and your country of origin. At this point all you can do is wait and be patient.

Sponsor I864

You can file the most recent tax return. You should have an attorney assist you.

If am a 21 yr US citizen can I petition for my father who entered with a tourist visa?

You can file for your father. However he needs to remain in status while his application is pending. It is best to have an attorney assist you. ...

Visiting my wife in the US while I-130 processing is pending: risks of denied entry?

You would need to have all the documents translated. It is probably best to remain in Italy and wait for your green card before coming to the US.

The immigration site says no appointments are available at any field office. What should I do? Hire a lawyer? Keep trying?

You should hire an immigration attorney. You should also look for offices that may be further away from you.

Stay after L1 and i94 expiry

You should consult your attorney. As long as the extension is filed while you are in status you should be okay.

What to do if i miss the interview's day?

Since it is over six years ago, it is likely that your father will have to reapply for her.