Can theology still be defined "the science of faith"?

To your question of whether theology—what St. Thomas Aquinas calls sacra doctrina ("sacred doctrine")—is possible, see

• Summa Theologica I q. 1 a. 2 "Whether sacred doctrine is a science?"

  Sacred doctrine is a science. We must bear in mind that there are two kinds of sciences. There are some which proceed from a principle known by the natural light of intelligence, such as arithmetic and geometry and the like. There are some which proceed from principles known by the light of a higher science: thus the science of perspective proceeds from principles established by geometry, and music from principles established by arithmetic. So it is that sacred doctrine is a science because it proceeds from principles established by the light of a higher science, namely, the science of God and the blessed. Hence, just as the musician accepts on authority the principles taught him by the mathematician, so sacred science is established on principles revealed by God.

To your questions "Is not science perhaps the opposite of faith?" and "Does not faith cease to be faith when it becomes science?," see

• Summa Theologica II-II q. 1 a. 5 "Whether those things that are of faith can be an object of science*?"
  ^{*["Science" (scientia) "is certain knowledge of a demonstrated conclusion through its demonstration."]}

  All science is derived from self-evident and therefore "seen" principles; wherefore all objects of science must needs be, in a fashion, seen.
  
  Now as stated above (Article [4]), it is impossible that one and the same thing should be believed and seen by the same person. Hence it is equally impossible for one and the same thing to be an object of science and of belief for the same person. It may happen, however, that a thing which is an object of vision or science for one, is believed by another: since we hope to see some day what we now believe about the Trinity, according to 1 Cor. 13:12: "We see now through a glass in a dark manner; but then face to face": which vision the angels possess already; so that what we believe, they see. In like manner it may happen that what is an object of vision or scientific knowledge for one man, even in the state of a wayfarer, is, for another man, an object of faith, because he does not know it by demonstration.
  
  Nevertheless that which is proposed to be believed equally by all, is equally unknown by all as an object of science: such are the things which are of faith simply. Consequently faith and science are not about the same things.

"3) Is not science perhaps the opposite of faith?"

I've seen again and again that people define faith as meaning to believe something apart from facts (or some variant). Yet when I read the use of faith in the Bible, I don't see that at all. For instance, Jesus inspired faith in Himself as the Messiah. How? By blind chance? No; the people expected the Messiah to come and do miracles, so He performed miracles -- "When the Christ comes, will He do more signs than these which this Man has done?" (John 7:31b). They believed because of the proof offered.

God has given us a great many prophecies concerning the coming of the Christ so we would have reason to believe that God sent Him, and Matthew records many of these prophecies to convince the Jews that Jesus was the Christ. He gave them reason to believe.

Faith in the Bible simply means to have a conviction that something is so, no more and no less. How we obtain that conviction is not found in the word. As we see in the Bible over and over, biblical conviction of the things of God include reasons to believe; God does not overlook the mind He made in us.

Yes, the term is applicable to theology. Let's break it down to see why.

The first word is science. At its root, the word simply means knowledge. Many people forget that and the word has been somewhat hijacked to mean exclusively study about the natural world derived from facts and experiments, or more properly, the scientific method. The scientific method is a science in its own, but it is not science in whole.

The second word is faith. First, the word simply means having complete trust in something. You can have faith in just about anything, and hopefully you would have a reason to do so. I can have faith in a person to do the right thing, which might be based on our shared experiences. I can have faith in the scientific method to provide the best current understanding about something, which might be based on the history of its use. And, of course, I can have faith in God, that he is there and cares about us, that he will keep his word, that Christ will return, and that eternal life awaits for those who love Him. This can be based on any number of things including, apologetics, being convinced the Gospels and New Testament are true, and personal revelation. Again, many people forget that the word has been hijacked to mean believe without reason, or what is colloquially called blind faith. The word simply does not mean that and that phrase and the actually definition of faith are contradictory. In short, faith means believing in what you have reason to believe.

So the phrase the science of faith means exactly knowledge of what there is reason to believe. Specifically, in this context, the author means exclusively items concerning Christianity and the Christian God. So alternatively, the author may mean knowledge of the Christian religion with assumptions that it is true at its core. Slightly different than what I have come up with, but acceptable and understandable considering the time and place that he said that.