When filling out an N400 application, if we don't list "other names used" will it speed up the background check process?

No, it won't speed it up. Failing to include other names could also constitute misrepresentation. Be fully truthful. You should consider consulting...

Can my child qualify for citizenship?

It depends on how old your child was at the time you became a US citizen. You should consider consulting with an experienced immigration attorney...

2nd Texas speeding ticket within 6 mos.

You can qualify for deferral. Talk to the prosecutor. Also hiring a local traffic defense attorney to represent you.

Why haven’t I received a NVC case number?

Contact USCIS or the NVC to see if they have your case. This may also depend on whether your priority date is current. It's advisable to discuss...

What can happen if the police didn't notice I was driving with the registration of another vehicle?

It's hard for anyone on a general forum to answer as to what a particular judge may or may not see. It's advisable to consult with a local traffic...

About marriage base case

He will need to prove that the marriage is bonafide and entered into with good faith, to get legal status based on your marriage.

The online system is unreliable, at best. Either call USCIS customer service to check status OR consult with an attorney who can look into it for you.

What steps do I take now for my GC Interview?

It's advisable to discuss this matter with your attorney or consult with another attorney who can review the file.

Work permit and SSN

No, the children cannot get a SSN or work permit. They can possibly apply for an ITIN, if they qualify.

Possibly. However, considering the focus on I-864 right now, its advisable to consult with an immigration attorney to review your documents, at least.