They do, but the current leadership does not have any appetite for entering Iran's airspace after the last downing drama. If Iran captures a pilot, there is going to be no more room for market manipulation.
I doubt the identity of the pilot & WSO will be released -- why would it -- at least not until they retire and write a book about it and out themselves.
Specific detail, we don't and won't have enough control to do aerial refueling in range of Iran. Tankers aren't stealthy or fast, I'm not sure if they have any countermeasures at all.
One can look at a map of strikes and draw the line on our range, the northeast half of the country is mostly untouched.