You filed for U.S. asylum in Mexico and then entered the U.S.? Or you entered without inspection and now you want to file for asylum? Certainly...

You will need consular processing to return to the U.S. on that visa.

It could be a problem if you failed to disclose it in your other applications.

You do not need to live in the U.S. at all to apply for a green card, as an EB1, or in any other category of immigrant visas.

It seems to me that the answer is yes, he was denied, and he was denied because he could not prove nonimmigrant intent.

This site is for U.S. immigration. You should contact a Canadian immigration lawyers with questions about Canadian immigration.

It means that they may be questioned when the seek to be readmitted to the U.S. as a J1, since perhaps they intend to change status.

Could be there is some issue, this is very unusual. If you are certain there is no issue you should file a lawsuit.

You would have to prove that you do not intend to immigrate to the U.S. and you may have trouble convincing the consular officer of that.

You do not need permission from the U.S. to work in your country. You are not in H1b status if you are not in the U.S.