Hey, we are all speakers of English as it is found in Scotland, so there are undoubtably small discrepancies with other varieties of English. If we changed it to “a toothache” we would be having the same conversion with Scottish users wishing it was the other way.
Tha an dèideadh air would be "he has (a/the) toothache", but there it's acting as a prepositional pronoun.
In tha an dèideadh air Calum, air is acting as the preposition "on", so it's "Calum has (a/the) toothache".
As to why it's air versus aig, things related to body parts tend to be air because aig tends to be actual possession.