You need a refugee travel document. Re-entry permit is if you will be out of the country for six months or longer (not recommended). From the wording of the question, it sounds as though Mexico is not the country from which you sought asylum.
Definitely do not file a new petition. Send a copy of the receipt notice, the baby's birth certificate, the baby's parents marriage certificate with a very short very clear letter identifying the petitioner, the beneficiary, the case number. Then state simply in one or two sentences that since filing the petition the beneficiary has had a baby, and you are adding the baby as a derivative. Give baby's name and date if birth. Lust attached docs. An attorney can help you if you like
Get a lawyer right away. I have been able to deal with similar situations in the past, but it totally depends on the specific facts of each individual case. You need an attorney to review everything in detail. Way more than you can on this forum, not to mention, you must do it in a manner that is confidential, which this public forum definitely is not.
Basically, yes, but there are things that need to be done. Also, I'm assuming you were refused entry but not actually removed. Those exact facts regarding what happened need to be clarified. An experienced immigration attorney can help alleviate a lot of your stress.
Only citizens can petition for children who are married, so either you'd have to go through your brother, or your dad needs to naturalize. Which way is faster depends in part on the country you are from. You can check in the Visa Bulletin for the current priority dates. The priority date is the date the petition is received. Someone with the priority date listed in the Visa Bulletin or earlier, can proceed with the case. Note it is not valid to look at the dates and conclude that since...
No. If your application was rejected, you should meet with an experienced immigration attorney right away, because there are time deadlines to do something about it. If you have not yet applied, note that you cannot be denied for being unemployed, but you do need to show you've been filing tax returns when required.
The issue would be the arrest. The use of fraudulent documents to obtain an immigration benefit would be a problem if it were to obtain a U.S. immigration benefit, which it wasn't, and the UK subsequently granted her citizenship, so that one shouldn't be a problem. However, the arrest needs to be evaluated. Obtain all the documentation regarding that case including the exact charge and citation to the section of law (so that an attorney can research it), as well as the sentence, and...
Seattle is very good on these issues, and really tries to be accommodating, but it does depend on their workload. As long as you have an appointment notice (even if for a different day), you can attempt to do it as a walk-in, but you should arrive extremely early, and be prepared to wait. If they can get you in, they will. An option is to reschedule the appointment.
When you receive the receipt notice, the top section will have a receipt number. The receipt number is the case number. If the person you are petitioning for will need to consular process abroad, when the petition is approved and transferred to the National Visa Center, it will be assigned another case number. That one will be on correspondence from the NVC. It would help to have an immigration attorney take you through the process
If the government is not opposed, the judge is very likely to grant it. I would continue to prepare, anyway. You want to submit all evidence and make a complete record for the future just in case. The judge can still deny it, but it would be extremely rare