Sure they can listen to it in their head if they have perfect pitch and can hear the pitches in their head without having to go to a piano, otherwise they would have to be at a piano for that to work. They need to be able to identify parallel fifths with just their eyes when they are writing 4-part harmony on a piece of paper.
Yes, that is a perfectly valid of working out an augmented sixth, assuming the student remembers that it sounds like a minor seventh.
I do not mean to suggest that the ways I suggest are the only ways, but I think it's important to be able to work out all intervals with whatever method you have. If the student can only identify perfect, major and minor intervals, but never augmented or diminished, there is an issue.
If you can find a major 6th, then a minor 6th is a semitone lower, and an aug6 is a semitone higher.
Yes, as I said the Wharram method is a perfectly fine method. You can find a major or minor sixth and make it larger by a semitone or make it smaller by a semitone to get another interval, and you just have to be able to reason out what that interval is.
It doesn't matter what technique they use as long as:
a) it doesn't leave a gap where they may be unable to identify an interval that they may encounter in actual music
b) it isn't such a slow technique that it takes them a long time to determine the quality of an interval