After the collapse of the Byzantine Empire, what happened to the Imperial Byzantine family? Did they later establish other smaller kingdoms? Do any European Royal family trace their ancestry to them (Especially in eastern Europe). Is there anyone in this time who holds the claim to throne of the empire (as in the case of many dissolved monarchies)?
Byzantine Empire was not formally a hereditary monarchy. There was no law which regulated inheritance in Byzantine Empire.
Nevertheless the offsprings of the imperial family sold the right to claim the throne to Ferdinand of Aragon and Isabella of Castile, Spanish monarchs. This was inherited by Charles V, Holy Roman emperor. Yet he never styled himself a Byzantine or Constantinopolian emperor. Being a Roman Emperor was a part of his title though after he was crowned Roman Emperor by the Pope (he was the last man to receive this title ever).
In addition to Anixx's answer, check out Andreas Palaiologos - the oldest newphew of Constantine (the last emperor). Looks like he sold his "rights" to the Byzantine throne twice, both to France and Spain. And his younger brother sold them to... gasp... the Ottomans. Well, they had to get some money for high living.
Which brings us to another aspect: Mehmed II claimed that he actually was the new Roman Emperor, or Kayser-i Rum as he called it, by right of conquest. (He did have a point there). I know that Suleiman the Magnificent called himself so as well (in a letter to Charles V) but wikipedia, linked to in "claimed" above, a ssertsthat later sultans dropped the title.
see my several answers to this question "Greek Revolution- where did the greeks look for descendants of the byzantine dynasty?"
A short answer is that the male line descendants died out in a few generations as far as is known, but several Palaiologos princesses married into other families and their descendants lasted after the last known male lineage descendants of the Palaiologos dynasty.
Some of the female line descndnts exist today. Thus going by male preference primogeniture the rightful heir of the Palaiologos dynasty is probably Luigi Serra, 11th Duke of Cassano.
yes, the head of the Imperial House lives in Moscow. He is H.I.H Prince Vladimir Gorshkov-Cantacuzene. check the website www.royal-byzantium.com