The books you're speaking of are the Biblical apocrypha. I'm going to limit myself to discussing the New Testament apocrypha in this answer, because if I included the Hebrew Bible apocrypha, this answer would be even longer.
 

So I vaguely remember a scene from a Tom Hanks film where they discuss a meeting in Constantinople (I think) where a bunch of folks got together and essentially kicked out books for being canon. Did this really happen?

What the book and film were referencing was the First Council of Nicaea in 325. However, Dan Brown was wrong about this point. It's a common myth that the biblical canon was decided at the Council of Nicaea that was most likely invented or at least popularized by Voltaire. In reality, the development of the New Testament canon took centuries. This aside, most of the Gnostic Gospels (discussed below) had already been attacked as heretical long before 325 by (among others) Irenaeus of Lyon and Tertullian.
 

Any reason for the books (overarching themes, ideologies or otherwise) that were removed

The most common reason for exclusion was that the theology was considered heretical.

This is the case for the so-called Gnostic Gospels that include the Apocryphon of John, the Gospel of Judas, the Gospel of Mary, the Secret Book of John, the Greek Gospel of the Egyptians, etc. Contrary to what The Da Vinci Code claims, these works are mostly products of the 2nd, 3rd and 4th centuries, so they are not older than or even contemporary to the Four Canonical Gospels. It's hard to talk about Christian Gnosticism with any authority because so little is known of it, but it seems to have mostly centered around the idea that the world was the creation of either Satan or an evil demiurge who was usually identified with YHWH, the God of the Hebrews.

Also excluded on the basis of heresy was the Gospel of the Lord, better known as the Gospel of Marcion. Marcion of Sinope was a 2nd century Christian whose beliefs were somewhat similar to Gnostics, in that he believed that YHWH was an evil demiurge, but his conception of the True God was different. At any rate, his teachings were attacked by some of his near-contemporaries who claimed that much of the Gospel of the Lord was actually a revision of the Gospel of Luke (and many modern scholars think that that charge was true). Another work considered heretical was the Gospel of Mani, a central holy book to the Manicheans. Unlike Marcionism and Gnosticism, Manichaeism can't really be considered a sect of Christianity; it was an entirely different religion that also incorporated elements of Zoroastrianism and Buddhism. Unfortunately, no complete copies of the Gospels of Marcion and Mani have survived.

A final book rejected on the basis of heresy was the Gospel of Peter, which condemned most likely because of its Docetist tendencies--Docetism was the idea that Jesus' human form was an illusion. It also lays the blame for the crucifixion of Jesus solely on Herod Antipas and the Jews.

Works might also much be considered apocryphal if they were attributed to someone who couldn't have possibly written them. This was the case for the Apocalypse of Pseudo-Methodius. It claims to be the work of the 4th century Methodius of Olympus, but it probably written in the late 7th century, after the conquests of the Rashidun and Umayyad Caliphates. It "prophesies" that the sons of Ishmael (that is, Arabs) will invade to punish Christians for their sinfulness but be driven out by a divine Roman emperor.

My own personal favorites of the apocryphal books are the Infancy Gospel of Thomas and the Apocalypse of Peter. The Infancy Gospel of Thomas is about the childhood of Jesus and mostly presents him as Dennis the Menace with magical powers. It was attacked because it may have been Gnostic in origin, but remained popular well into the Middle Ages. The bits of the Apocalypse of Peter that survive mainly discuss Heaven and Hell and I find them interesting because they are somewhat similar to the heaven and hell that Dante Alighieri wrote of. Unlike most of the works I’ve mentioned, the Apocalypse of Peter came close to being recognized as orthodox, as did The Shepherd of Hermas and the Epistle of Barnabas. On the other hand, the Book of Revelation was the last book that was declared to be canonical.
 

and are any of them available to be studied today

Not only that, but many are available (in English translation and sometimes in their original language as well) for free on the internet.