I'd like to break down your questions into parts
My question is this: With the abundance of Biblical passages, in both the Old and New Testament, how do Christians rationalize the specific stress and emphasis on only certain sections of the Bible?
It is understood that the laws laid out in the old testament were part of the 'old covenant' or the old promise and that God through Jesus has made a 'new covenant' with mankind.
Peter, in the book of Acts, see a vision of many of the foods that it is forbidden for a Jewish person to eat, and hears God telling him to eat it. Afterwards he is invited into the home of a Roman general, which pious Jewish person would never enter. Peter understand that this is God lifting many of the restrictions on Jewish law. This is later formalized in Acts.
For instance, people often quote Leviticus on denouncing homosexuality, and yet Leviticus also writes against eating shellfish and wearing cotton/polyester blend clothing
This is one of the sins that Paul mentioned explicitly in the new testament in a couple of places (see The Preacher's answer). I am not sure why people quote the O.T. as much as the N.T.
Judeo-Christian stance on women and slavery has changed over the past few millennial.
There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus.
This is a RADICAL shift in thinking about the nature of people's place in society, in particular the treatment of women and slaves. I believe that it has taken us this long to even get our heads around this concept and accept is as a larger concept (in the west). We still have a long way to go, if we really get this notion ingrained in us the world would be unimaginably better, closer to what God intended for us.
Beyond that, the Bible has been translated by hand between multiple languages, and been reviewed, revised, cut, and supplemented by several religious authorities. If the entire Bible is the infallible word of God, how can his followers assert their own judgement over his by themselves choosing which parts of the Bible are true or applicable?
It should be noted that their are many Christians who do not believe that the Bible is infallible. The infallibility doctrine is a very prominent belief among evangelicals in the United States, but please do not think this is necessarily the same among ALL Christians, especially those that would consider themselves 'Liberal'.
I think it is dangerous to rest your entire faith on something like the infallibility doctrine. Because many, when they go to college or investigate themselves learn things like, for instance the story of the adulterous woman in John, is probably a later addition to the text, and this shakes their faith.
how can his followers assert their own judgement over his by themselves choosing which parts of the Bible are true or applicable?
Jesus said in Matthew 7:7
Ask and it will be given to you; seek and you will find; knock and the door will be opened to you.
If you read the text as it was intended you will find a consistent message of a loving God, who is asking us to seek after him in righteousness. I believe that if you earnestly seek after him, as Jesus said you will find what you are looking for.