Vermax seems to be an interesting case, though, since we're not entirely sure how it went down, only that it crashed into the water and took Jacaerys with it after he was shot by crossbow fire. Odds are the dragon entered too steep a dive and built up too much momentum to be able to pull up in time, crashing into the ground, but that's not the kind of mistake an older dragon like Caraxes would make.
A scorpion bolt has the force and speed to punch through that flesh and hit the vital parts behind, hence how the Dornish killed Meraxes - the hit penetrated the eye and hit the brain directly, resulting in instant death. An arrow, not even one fired from something like a dragonbone longbow, would have the punching power needed to get through so much soft tissue in a single hit.
In those cases, it was a siege weapon that did the job - the scorpion. An arrow, like your archer would have to be using, wouldn't actually have the penetrating power needed for a lethal hit - we know in universe that dragons don't die all that easily even if you shoot down the throat, and an arrow to the eye would likely blind it in one eye but get lodged in the flesh before it reached the brain.
In the air, things get even harder if you're going with small weapons - a dragon, despite its size, is going to be travelling at *quite* a speed, especially on a dive, but it's not impossible. We've got historical information in setting about the Dornish offing Meraxes and the Three Daughters doing equally well during the DOD.
The vestigial claws on the wings and the hind legs, however, are likely not all that dangerous - more so in an air battle, I think, when the dragon would be able to grapple its opponent. The real danger there would be the sheer concussive force of the impact since it would likely be able to crush bones, and the tail could possibly do such a thing as well.
Even then, odds are the flesh melt and will fuse with the steel, meaning that burns that could be survived with the medical knowledge of Westeros will still be lethal due to the following infection. The dragon's jaws are similarly dangerous, and could likely decapitate a man in armor and his own horse as well.
...the heat of a dragon's breath on a near miss is nearly as dangerous as it is on a direct hit, since the individual inside won't be able to cool down as effectively and will quickly develop heat stroke. A direct hit is likely to be mortal, even if the death is slow or takes several hours.
The direct heat of the flame is obviously one problem, but another that is rarely considered is the oxygen that the fire will take out of the air in the process of burning - an attacker could very well find himself suffocated by the smoke of the burning grass or by the lack of breathable air, as well as being cooked inside his armor. In the tight confines of full plate armor where there is little ventilation...