Last semester I took a grad class on the Crusades, but I don't have my books on me right now so I'll cite my sources later. The short answer on both fronts is: Yes. The Crusaders did pay homage to Alexios I on their way to the Levant. However, after the Battle of Antioch the Byzantine contingent that was sent to aid the Crusaders, which was led by Alexios' general Tatikios, fled from the battle. After the Crusaders were successful in their acquisition of the city they felt that since Alexios had failed to support his end of the vassalage contract by supporting the Crusading army (since Tatikios retreated) their agreement was no longer binding.
However, the Byzantine Empire was able to reclaim much of Anatolia which had been conquered by the Seljiuq Turks (which is what spurred Alexios to seek aid from Pope Urban II in the first place). In addition to this, Bohemund, who had named himself Prince of Antioch after the siege, went to war with Alexios briefly in 1107. Out of this war came the Treaty of Devol in 1108 which made the Principality of Antioch a vassal state of the Byzantine Empire, where it served as a buffer against Muslim incursion.
Here are some Wikipedia articles to check out until I get my print sources.
EDIT: Print Sources
The Crusades: A Reader (Readings in Medieval Civilizations and Cultures)- edited by S.J. Allen and Emilie Amt
The Crusades: The Authoritative History of the War for the Holy Land by Thomas Asbridge
The Byzantines, with the help of the crusaders, managed to reclaim much of Anatolia from the Seljuks.
