His being called by his first name is not surprising; think of
Leonardo,
Michelangelo,
Rembrandt.
The Nose
When he was 20, he got into a duel with another man over who was the better mathematician (or perhaps over an arcane mathematical dispute); Tycho came out of it minus the bridge of his nose.
His prosthetic nose was traditionally said to have been made of a silver-gold alloy, but according to thestraightdope.com, when his tomb was opened in 1901, the nasal opening of the skull was tinted green -- a sign of exposure to copper. (Either Tycho himself was cheated by his nosemaker and thought the nose was silver-gold, or else he had it made of copper and coated with silver-gold to more closely approximate skin color.)
As for fastening methods, contemporary accounts describe him carrying the nose in a special tin of goo, either to polish it or to stick it to his face. There's no mention of straps or hooks or any other method of attaching the false nose to his face.
Other Weird Facts
His life was extraordinary, as he was an eccentric, belligerent, tyrannical, pompous man who had tremendous wealth and prestige. He received extremely generous patronage from King Frederick II of Denmark and Norway, in the form of about $5 billion, a private island (Hven), and licence to do as he pleased. On the island, he built his "castle of the heavens" (Uraniborg), which had a state-of-the-art astronomical observatory and room for lots of mischief. The castle became a scientific center on par with a modern university, but it was not without its idiosyncrasies; Tycho had a dungeon built in the castle to imprison visiting scholars who displeased him. He kept a dwarf named Jeppe as a servant/jester*, subjecting him to very degrading treatment -- Jeppe slept at the foot of Tycho's bed, had to beg for table scraps, etc. Tycho had a pet elk, which roamed the castle freely and finally died while falling down some stairs drunk. He had eight illegitimate children with a peasant woman (Kirstine). After the king died, also of drinking too much, Tycho's free ride became rockier; he was driven from Denmark and settled in Prague, where he finally died of a burst bladder.
* This may not be true. One modern account I've seen says the dwarf-jester's name was Per Gek, and he was only with Tycho for a few months. But it's too good a story to pass up.
This is only the dirt, so I've said nothing of his science. A clear, concise web biography, with good explanations of his scientific importance, is at es.rice.edu/ES/humsoc/Galileo/People/tycho_brahe.html. According to that page, the standard full biography is Victor E. Thoren's The Lord of Uraniborg: A Biography of Tycho Brahe (Cambridge: Cambridge University Press, 1990).