The theologian Thomas Aquinas wrote much in his brief forty-nine years of life. Many people are introduced to him through a brief passage in his Summa theologiae in which he proposes five possible "ways" for proving God's existence. In various manners, these five proofs are based on ways that someone can start with human experience and prove from that finite, changing experience how there must be an unchanging God. 

Many of the other thinkers listed in this question as potential answers had interest in matters similar to this as well. Most directly pertinent for this matter is Anselm of Canterbury. Anselm is well known for his so-called "ontological argument" for God's existence (though the title is a bit of a later attribution). His general idea was that so long as you can have an idea of, "Something than which nothing greater can be thought," you can prove that such a great thing must indeed exist—precisely because it is so great that it is perfect and hence has the perfection of existence.

Interestingly, many of the works of Aristotle returned to the Latin West by means of Arabic translations. Often the Latin editions were Latin translations of Arabic translations of Syriac translations of the Greek. Eventually, better Greek editions came to the West, but the Islamic commentators on Aristotle remained very important.

The two greatest figures (though there were others) were Ibn Rushd (or Averroes) and Ibn Sina (or Avicenna). Many of the works of Aristotle were very difficult to read. Part of this was due to the difficulty of the texts but part of it was also due to the poor translations (which were often very literalistic). Ibn Rushd had written a number of commentaries on Aristotle's works. Latin translations of these commentaries were often consulted by medieval philosophers so as to help in the parsing of Aristotle's texts. (Indeed, in some editions of Aristotle's works, Ibn Rushd's commentaries were included in the margins.)

Ibn Sina was a unique and creative Persian philosopher who combined Aristotle's thought with many other forms of philosophical thought (especially those of Neo-Platonism). Ibn Rushd thought that Ibn Sina's thought was not faithful to Aristotle. Nevertheless, Ibn Sina's writings on natural philosophy and metaphysics (or "first philosophy") were very important for medieval thinkers who were learning anew the details of Aristotle's thought.

The Middle Ages was a variegated and differentiated period, filled with many thinkers of quite differing temperaments and ideas. The general theme that united the period, at least in its time of greatest development, was the advent of the universities and the eventual division of camps of thinkers into various "schools"—i.e. schools of thought such as Thomism, Nominalism, Scotism, Albertism, and so forth. For this reason, the period is often identified with the title "Scholasticism," though you should always remember that this one term really applies to many different perspectives and outlooks.

By the 13th and 14th century, the liberal arts curriculum had generally stabilized and arrived at what became its classical formulation. These topics were studied by all before going on to higher studies such as theology, law, or medicine. Also, during this same period, other topics were added to the arts, especially natural philosophy but also a study of other works of Aristotle such as the Metaphysics and the Nicomachean Ethics.

The liberal arts were split up into two main branches. The first consisted of three subjects pertaining to logical expression, while the second consisted of four subjects pertaining to mathematical matters. The first three were called the "trivium", literally meaning "the three subjects": grammar, logic, and rhetoric. The latter were called the "quadrivium", literally meaning "the four subjects": arithmetic, geometry, astronomy, and music (understood as a science of mathematical proportions).

The Sentences of Peter Lombard (1096-1160) was the single most influential textbook for many, many centuries. It was a compilation of sources, systematized for the study of theology. In the universities, it became the custom that the "dissertation"—it really was not that, strictly speaking—required the writing of a commentary on all four books of Lombard's Sentences. These works were often massive and involve a great deal of philosophical and theological speculation. Often these commentaries involved thousands upon thousands of pages. This was a requirement of university students for centuries—even Martin Luther wrote a commentary on this work, as did other students of his time!