What is the style of writing used in the Summa Theologica called?

I think the word you are looking for is quodlibet or quodlibetal. This comes from Latin words meaning "whatever you want".

The Wikipedia entry for Scholasticism explains

Scholastic schools had two methods of teaching. The first was the lectio: a teacher would read a text, expounding on certain words and ideas, but no questions were permitted; it was a simple reading of a text: instructors explained, and students listened in silence.

The second was the disputatio, which goes right to the heart of scholasticism. There were two types of disputationes: the first was the "ordinary" type, whereby the questions (quaestiones) to be disputed were announced beforehand; the second was the quodlibetal, whereby the students proposed a question to the teacher without prior preparation. The teacher advanced a response, citing authoritative texts to prove his position. Students then rebutted the response, and the quodlibetal went back and forth. Someone took notes on what was said, allowing the teacher to summarise all arguments and present his final position the following day, riposting all rebuttals.

The term is now used in music; OED has the meaning as "A musical composition combining several different melodies (usually popular tunes) in counterpoint, often in a light-hearted manner." It says there that the academic "quodlibet" originated at the University of Paris in the 1230s; Aquinas lived from 1225 to 1274.

The following is from a note on Wikipedia:

"St. Thomas Aquinas used the "Grand Commentary" of Averroes as his model, being, apparently, the first Scholastic to adopt that style of exposition..." — Turner, William. "Averroes." The Catholic Encyclopedia. Vol. 2. New York: Robert Appleton Company (1907).

The name of the style is "disputation". "Quodlibetal" is the adjective applied when the disputation could be about anything.