It is not a true statement to say that all Christians believe in the young Earth theory – that dinosaurs and people co-existed, the earth is only 6000 years old, and so forth. Many sincere believers hold differing views on the account given in Genesis, and there is not enough information to establish a “my-way-or-else” dogma on the subject (I used to be one of those).
That said, you bring up several interesting points. First, to the question of the dinosaurs. It is important to remember that, if you take a strict literal interpretation of Genesis, until Adam and Eve sinned, death was apparently not part of the original design, in which case the T-Rex in the garden was as tame as the lamb. Also, recall that God created the man and the woman to have dominion over the earth and all its creatures. God granting dominion would have included at least His protection of the new governors.
The extinction of the dinosaurs poses no real problem to literalists because one can easily imagine a scenario in which the two dinosaurs of a given species fail to adapt to the new climate and die shortly after the Flood. Think how many species have been driven to extinction without a catastrophic event, and you start to realize that many of the survivors of the flood were as likely as not wiped out shortly thereafter. As to the question of the ark, one can assume that young animals were brought on board rather than full grown. This would make it easier to feed and care for the animals, as well as making them safer to handle while using less space. Also, the Bible says that God commanded Noah to bring two of every KIND of animal, not necessarily two of every species. It may have been enough to bring two dogs, an not worry much as to whether they were hounds or terriers. Much of the present diversity among animals can be traced to a handful of types that have adapted to different environments and conditions. Noah worked on the ark for 120 years (lots of theories on the length of life in those days, too many to get in to here), and one can assume that he and his sons started gathering the animals early on. Remember also that God commanded the animals to be found, so He likely guided Noah and his family to locate the kinds He commanded.
Now to the race question – I’m sure the overwhelming majority of Christians believe in the fact of genetics. It is not true that Adam and Eve were white. The Bible doesn’t say what race they were, but it is interesting to note that Adam is derived from a Hebrew word that means roughly “son of the red earth.” Assuming that mankind started as not white, but some other color, it is easy enough to develop a wide range of variations fairly quickly. Modern evolutionary science tends to believe that all mankind came from one original ancestor or ancestor group of the same or similar racial stock, and that we adapted different skin tones to adapt to different climates, etc. (Someone more knowledgeable on this subject may feel free to correct my oversimplified generalities.)
Again, these are only speculations and not iron clad. What the Genesis account tells us above all is that mankind is the special creation of God, the crowning achievement of all He made. We are not here by accident or mistake. God took a person interest in making us, and even after being rejected by man, God pursued a relationship with men and women. Genesis is a testimony of God’s power, goodness, mercy, and love for all people.