I think a distinction should be made between racial and ethnic. Let's assume the word racial refers to phenotypes and ethnic refers to identity.

The ethnic composition of modern Turkey is hugely different. It went from a Greek speaking Christian majority to a Turkish speaking Muslim majority. So ethnically, there was a huge change from Byzantine to Modern Turkey. I'm not sure what the Armenian, Kurdish and Syriac populations were like in the Byzantine era.

However the racial/phenotype change isn't as significant. Genetic tests suggest that the bulk of modern Turkish peoples ancestry is from the pre-Turkic populations of Anatolia. However there is still a small but significant amount of Central Asian Turkic ancestry as well.

Anecdotally I can tell you that it's normal to see Turks with slight "Oriental" facial features. Indeed I find that many Eurasian (part European, part East Asian) people or part European part Native American etc. people tend to look Turkish to me. It's more common than people think and if you pay attention to the facial features of Turkish people you'll notice it. However in some cases it might just be pre-Turkic Anatolian features that coincidentally look Oriental.

So the answer is for the most part the modern Turkish population is a continuation of Byzantine Anatolians, except with the addition of a Turkic substrate to the genetics as well.

One must also keep in mind that migrations to Anatolia from Central Asia didn't just finish after the initial Turkish conquest of Anatolia, but continued over the centuries and there were also non-Anatolian Turkic peoples who were settled around the non-Anatolian Black Sea coast that immigrated to Anatolia as well. So any "Oriental" features you see on a Turkish person is not necessarily from the initial Seljuk migrations, but could be more recent than that.

Another thing to keep in mind is that many Turks can trace their ancestry back to the Balkans and Caucasus, since millions of Muslims from these regions, Turkish speaking and non-Turkish speaking, migrated to the remaining lands of the Ottoman Empire after fearing and receiving persecution when the Ottomans lost control of these territories. So with that in mind, in addition to the Balkan/Caucasus Turkish and Turkic influence, you get mainland and island Greek, Slavic, Albanian, Romanian, Hungarian, Circassian, Georgian and other Caucasus groups influence on the modern Turkish people as well.

Source: http://en.wikipedia.org/wiki/Genetic_history_of_the_Turkish_people

Yes.

Byzantine Anatolia was mostly orthodox christian and Greek or Armenian before the battle of Manzikert 1071. After that, Turkic people started migrating to Anatolia and mixing with the local population. This is an oddity in the history of nomad conquests that the nomads assimilated the local peasants rather than the other way around.

The Ottomans were usually not interested in making their christian subjects muslims or force them to speak Turkish - the christians of Anatolia had their own laws and court system and paid Jizya tax (an extra tax which allowed them to practice their faith but also exempted them from military service) either in form of money or by giving up a male child to the Janissaries. Right up to the 20th century, the Ottomans considered themselves Ottoman rather than Turkish (Turkish was mostly used as a slur akin to redneck - it meant someone uneducated and unsophisticated from the countryside). That said, there were conversions, especially as some christians wanted to avoid the Jizya tax.

Still, Anatolia still held large minorities (and in some regions, such as around Smyrna and in Pontus pluralities or even majorities) of Greeks until the population exchange (read forced expulsion) after the Greco-Turkish war of 1919-22. While a few Greeks remained, they were forced to learn and use Turkish and become fully assimilated into a nationalist Turkish (rather than Ottoman) state.