"1) Suppose a person does not have a valid H1B stamping in his passport (say, the old H1B stamp expired and his new stamping attempt at a consulate was a failure ). He has valid advance Parole. In port of entry can the person say that he has valid H1B I-797 and would like to work on H1B rather than EAD?
AT the POE they are looking for valid documents. Even if you enter using AP, this does not invalidate your H1B. I did same thrice and have no issues. Only once an officer asked whether I am still with the same company for which I replied yes.
Will the Port of Entry people cancel the I-797 and give an I94 based on Advance Parole? They do not cancel your I-797 as entering using AP does not invalidate it. They will issue a new I-94 and mark it as 'Pending AOS', if you enter using AP.
After doing that can the person get a new I-94 while extending the same I-797 in the future to get a new I-94? Very much yes. I did traveled thrice and extended my H1 after each time with the same employer
2) Suppose a person has a valid H1B visa stamped on his passport as well as a valid Advance Parole.
While entering the country through a port of entry if the person shows only the valid H1B visa, the Port of entry computer system will prompt the officer to ask the Advance Parole from the H1B person. The officer will say, "You can work in H1B, but u cannot hide AP". Show your AP. Is this true?
If you have valid visa and enter using valid H1, I do not think they request your to provide AP. They can view all transactions related to you and very well know that you have a valid AP, EAD based on pending I-485."