Mankar was dragonborn.
http://imperial-library.info/content/oblivion-mythic-dawn-commentaries-1
Yes, but again, so was the Champion of Cyrodiil, and (s)he couldn't wear it. I don't think Mankar being a Dragonborn could explain how he got to wear the Amulet. I was under the impression that only the moral (or strongest) guy related to Talos at one time could wear it. That is, not any Dragonborn, only the most fitting at a time. Then the next one after that one's died or dethroned, etc.
Difficult to say. We know for a fact that not all people that wore the amulet of Kings were related to previous wearers of the Amulet of Kings... It would seem incredibly unlikely that every warlord out there that ever managed conquer Cyrodiil and be proclaimed Emperor who could thus wear the amulet of Kings was also a dragonborn, I mean what are the odds?
I actually checked and there seems to be no Emperors not related to Talos at all; everyone's his/her predecessor's half-brother or distant cousin at the very least, even to the point where races change every now and then... But The Third Era Timeline only becomes vague somewhere around Camoran Usurper's time (who didn't actually reign as an Emperor nor use the Amulet himself, I bet. And the fires were lit all the time according to Oblivion, so there must've been an Emperor, I thinkm Cephorus II).
The Timeline doesn't tell how Cephorus II and Uriel V are related, but it's obvious they are since the later Uriels all could wear the Amulet. Every other relation is explicit.
Or maybe they traditionally didn't don it until after the fires were a' roarin'. Nothing points to the ascendant emperors having to wear the AoK in order to the light dragonfires.
Well Martin at least couldn't light the fires without the Amulet; the flames couldn't be "a-roarin'" without the Amulet.
Or do you state that Martin actually could have lit them without AoK, but he just presumed he couldn't? If that's true, the irony!