#asshai

2. Per GRRM, Targaryens are not immune to fire, but they do have some heat resistance, and enjoy things like hot baths and hot weather. The only one actually immune to fire was Dany, and specifically only during the miracle of her dragons' birth. (During her taming of Drogon right before she rode him the first time, she received burns on her hands.) Many Targaryens have died or been injured by fire, including Viserys Targaryen (Dany's brother and his "golden crown"), Aerion Targaryen (he drank wildfire because he thought it would turn him into a dragon. It didn't), Rhaenys Targaryen the Queen Who Never Was, Daeron "the Daring" Targaryen, Aegon II Targaryen, and Rhaenyra Targaryen.

As for illness, the "Doctrine of Exceptionalism" was a religious precept that King Jaehaerys I worked out with the Faith of the Seven, to give the Targaryens an exception on the Faith's anti-incest stance. It stated that Targaryens were different, exceptional, special people, closer to gods than men, because of their unique silver-gold hair and purple eyes, because they alone rode dragons, and because they never got sick. "There was fire in the blood of the dragon, it was reasoned, a purifying fire that burned out all such plagues." However, only a few years after Jaehaerys made this agreement (and married his sister Alysanne), their 7-year-old daughter Daenerys died of the Shivers, a severe-flu-like epidemic. (This put great doubt in their heart, but did anyone do anything about it? lol no.) Their daughter Maegelle later died of greyscale, and their son Baelon died of appendicitis. Later Targaryens have died of other epidemics, of the pox, of tuberculosis, and other diseases.

However -- some Targaryens have shown surprising resistance to illness. Aegon III sat with many victims of the Winter Fever epidemic, and never showed any symptoms. Dany herself cannot recall ever getting sick. (She is not immune to being poisoned, though.) There may be something specifically connected to being a dragonrider (though Baelon was one), or more specifically being a potential Prince That Was Promised? Again, this is connected to the greater mysteries of ASOIAF, to be resolved later.

3 & 4. The base of the Hightower -- not the tower itself, but its first level -- is an ancient fortress made of fused black stone, which is similar to Valyrian construction made by melting stone with dragonfire (such as the castle of Dragonstone, the walls of Volantis, and the Valyrian roads). However, it predates the Valyrian empire by millennia, and is plain without decoration, unlike how the Valyrians would twist the melted stone into artistic forms. There are also other ancient structures in the world, the Five Forts on the eastern border of Yi Ti, that are also made of this fused black stone in this plain style. Some maesters also think the Hightower fortress's labyrinthine design is similar to the Mazes of Lorath, also ancient structures, made by a vanished giant not-quite-human species (called the Mazemakers) in pre-history. GRRM has said "there were dragons everywhere, once" (there are indeed records of dragons in Westeros before Valyria, and dragon bones found in far distant places in the world) and the truly ancient Asshai'i histories claim to have taught the Valyrians the secrets of dragons, so there's a theory that there was a dragonriding culture long before the Valyrians who left behind these fused black stone structures. (More on this in the next answer, and you can see an older theory post of mine on the subject here. Also note I am certain this culture was not the Great Empire of the Dawn, they're unrelated.) One more great mystery!

5. Yeeahh... this may be true. Or it might not be. Septon Barth (Jaehaerys's Hand of the King, and a great researcher into the origins of dragons, with theories that made maesters call him crazy and the Faith burn his books) apparently theorized in his Unnatural History that the Valyrian dragons may have been created via bloodmagic, possibly by breeding wyverns (flying reptiles that do not breathe fire), possibly with firewyrms (wingless/legless earth-boring creatures that do breathe fire). There's also (as I said above) Valyrian legends that claim they found dragons nesting in the Fourteen Flames, but ancient texts from Asshai claim that dragons first came from the Shadow (the mountains around Asshai), and an ancient nameless people brought them to Valyria and taught the Valyrians the magic needed to control them. And there's a myth from Qarth that there used to be another moon that cracked open like an egg and millions of dragons came out. We do not yet know the true answer.

GRRM recently said "Septon Barth got most of it right", but what is "most"? Was there an incredibly ancient vanished species of dragons that the original Valyrians re-created? Did these Valyrians somehow breed these new dragons with themselves to make them easier to control? What we do know is that occasionally Targaryens have had monstrous dragon-like stillbirths. We do know that very rarely a dragon egg has hatched a "broken thing" that dies quickly, or a monstrous wingless wyrm that attacks its cradlemate, with no known reason why. We know that in the ruins of Valyria since its Doom, there are apparently mutated creatures that can lay eggs containing "worms with faces" and "snakes with hands" in human flesh, a horrific experience witnessed by Septon Barth that sent him on his path. It's a great great mystery, and there will apparently be an answer one day.