Could a old caution from 8 years ago show up on background check at my green card interview

What does an old caution mean? Were you arrested in the UK?

Stapling is fine, USCIS will find them and deposit them.

Just wondering does starting their green card application affect their entry to the U.S. if they have a valid visit visa?

You should simply consular process their green cards to avoid this potential trouble.

Will I be able to travel back to the US if I explain the situation as everything is completely legal?

The officer won't deport you, he will refuse you admission. Why not simply file the I-130 and consular process?

I wanted to have information as to what would i need to provide at this time with the i130 petition?

You should submit that again as well as affidavits from your mom and dad.

How & when I can apply for USA Green card lottery program and what could be the cost for it? What are eligibility criteria?

You can find the answers to most of your questions here:...

Can I file form I-130 for CR-1 Visa to the U.S. Consulate in Rio de Janeiro?

You will likely be required to file the Form at the USCIS Chicago Lockbox.

Trying to understand is H1B clock reset with the new H1B application date or the day we enter into US again?

What exactly is your question? Are you currently in the U.S. in H-1B status?

Can I apply for a B-2 to visit my US born son, and wait for him to petition a I-130 for me?

Have you ever received a B visa before? Sometimes they can be difficult to obtain if the applicant has U.S. citizen children.

Will they call me for an interview before I depart the country?

What case are you referencing that you looked up online?