Only by conquest. He assumed the Purple in the Eastern (read: Byzantine) Empire and went on to conquer Rome, which by that time had been in Ostrogoth hands for some 60 years.
No, not at all. Odoacer (the fellow who deposed Romulus Augustulus--the last Western Roman emperor) was officially a "viceroy" in Italy. He apparently sent the imperial regalia back to Constantinople and the Eastern emperor at the time confirmed him as Magister Militum or some such inferior office. Odoacer himself took the title "rex" of Italy, which was seen as inferior the Emperor in Constantinople.
Similarly, when Odoacer had a falling out with Constantinople, the Emperor Zeno sent Theodoric and his Ostrogoths to kick @ss and take names--which Theodoric promptly did. He then assumed the title of "rex" which Odoacer had held and was again, officially the viceroy of the Emperor in Constantinople--to the extent that he issued no laws in his own name without mentioning the emperor first (to my best recollection) and respected the rights of Roman citizens in Italy to live under Roman laws. In reality, he was a king, and there was very little the Emperor could do to keep him under control--short of sending another barbarian army to Italy--until, of course, Justinian and Belisarius came on the scene.
I doubt that the Eastern emperors normally thought of themselves as "Roman emperors", since Rome was in the Western empire. I'm not sure how the word "Byzantine" came into vogue to describe the East.
You bet they did. "Byzantine" is a much later (as in post-Renaissance) formulation. Here's the preamble to Justinian's Institutes (the Laws he enacted during his reign). Decide for yourself what he thought:
PREAMBLE OF THE INSTITUTES OR ELEMENTS OF OUR LORD THE MOST HOLY EMPEROR JUSTINIAN IN THE NAME OF OUR LORD JESUS CHRIST.
The Emperor Cæsar, Flavius, Justinianus, Alemannicus, Gothicus, Francicus, Germanicus, Anticus, Alanicus, Vandalicus, Africanus, Pious, Happy, Renowned, Victor and Triumpher, ever Augustus, to the Youth desirous of learning the laws.
It is expedient that the Imperial Majesty not only be distinguished by arms, but also be protected by laws, so that government may be justly administered in time of both war and peace, and the Roman Sovereign not only may emerge victorious from battle with the enemy, but also by legitimate measures may defeat the evil designs of wicked men and appear as strict in the administration of justice as triumphant over conquered foes.
(1) This twofold task We have now accomplished, by means of the greatest attention and care, and with the assistance of God. For barbarous nations, subjected to Our authority, acknowledge Our warlike exploits, and Africa, as well as other numerous provinces after so long a period of time have submitted to the Roman domination, and have again become a portion of Our Empire by means of Our conquests through the aid of Celestial Power, and all peoples in fact, are now governed by laws either promulgated or compiled by Us....
If you still have doubts, the full text of Justinian's Code in English may be found here:
http://www.constitution.org/sps/sps.htm