Here is what St Thomas Aquinas has to say on the subject of the Creation of the Angels:

61. THE CREATION OF THE ANGELS

62. Angels are creatures. They exist, not by necessity, but by having existence given to them. That is, they have existence by participation. Now, what has existence by participation receives this existence from that which has existence by its own essence. Only God exists by his own essence. Therefore, angels have their existence from God; they are created.

63. God alone exists from eternity. He creates things by producing them from nothing. Creatures exist after they were nonexistent. Hence angels do not exist from eternity.

64. It seems most likely that angels and the bodily world were created at the same time, not angels first (as a kind of independent world of spirits) and the bodily world afterwards. Angels are part of the universe, and no part is perfect if it be entirely severed from the whole, the totality, to which it belongs.

65. The angels were created in heaven. And it is fitting that creatures of the most perfect nature should be created in the most noble place.

Since the angels were created at one time, it stands to reason that God is not continuing to create more angel.

Short Answer to "does God create new angels?"

We don't know. Angels are created by God, and current teaching is that God has already created Angels. I can see nothing that would prevent further creation, if God so wills it, but the official teachings don't address that.

Discussion

The language in the Catechism (current teaching) holds that the angels were created and are assigned tasks as the will of the Creator, or the Son, is manifested. It does not mention further procreation, nor supplemental creation, of "more angels." Since the nature of the angel is "spirit" it's reasonable to conclude that God won't ever run out of angels. The CCC does not support the idea of angels being created by man, as held in your question.

291 "In the beginning was the Word. . . and the Word was God. . . all things were made through him, and without him was not anything made that was made." The New Testament reveals that God created everything by the eternal Word, his beloved Son. In him "all things were created, in heaven and on earth.. . all things were created through him and for him. He is before all things, and in him all things hold together." The Church's faith likewise confesses the creative action of the Holy Spirit, the "giver of life", "the Creator Spirit" (Veni, Creator Spiritus), the "source of every good".

All of creation is God's work. Angel's are a part of creation.

332 Angels have been present since creation and throughout the history of salvation, announcing this salvation from afar or near and serving the accomplishment of the divine plan: they closed the earthly paradise; protected Lot; saved Hagar and her child; stayed Abraham's hand; communicated the law by their ministry; led the People of God; announced births and callings; and assisted the prophets, just to cite a few examples.

The current teaching reaches back to St Augustine to add depth to its position on Angels.

329 St. Augustine says: "'Angel' is the name of their office, not of their nature. If you seek the name of their nature, it is 'spirit'; if you seek the name of their office, it is 'angel': from what they are, 'spirit', from what they do, 'angel.' "With their whole beings the angels are servants and messengers of God. Because they "always behold the face of my Father who is in heaven" they are the "mighty ones who do his word, hearkening to the voice of his word". {refers to St. Augustine, and further to Matthew 18:10 and Psalm 103}
330 As purely spiritual creatures angels have intelligence and will: they are personal and immortal creatures, surpassing in perfection all visible creatures, as the splendour of their glory bears witness. {reference Pius XII, Humani generis: DS 3891; also Luke 20:36; Daniel 10:9- 12}

A variety of tasks can be assigned to these messengers from God, which includes keeping a watchful eye on mortals.

CCC 336 From infancy to death human life is surrounded by their watchful care and intercession. "Beside each believer stands an angel as protector and shepherd leading him to life."

To sum up: God created angels, there isn't anything in current teaching to support God needing any more, nor any teaching or belief that God won't or can't. There may be previous teachings and traditions that allude to that.

Were God to choose to create more angels, their being created beings and God being the creator, more angels would arrive. The current teachings look to be silent on that, probably finding this a question that is unnecessary to ask. "With God, all things are possible" covers a lot of ground.

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A few additional points, you may find the entry at the Catholic Encyclopedia, on Angels, of interest in tying OT and NT references to angels, and even older Semitic understanding of angels. From the Catholic Encyclopedia, which does not AFAIK have the stamp of doctrine or official teaching ...

(Latin angelus; Greek aggelos; from the Hebrew for "one going" or "one sent"; messenger). The word is used in Hebrew to denote indifferently either a divine or human messenger. The Septuagint renders it by aggelos which also has both significations. The Latin version, however, distinguishes the divine or spirit-messenger from the human, rendering the original in the one case by angelus and in the other by legatus or more generally by nuntius. In a few passages the Latin version is misleading, the word angelus being used where nuntius would have better expressed the meaning, e.g. Isaiah 18:2; 33:3-6.

From the "Treatise on the Angels" in St. Thomas Aquinas's Summa Theologica (q 61. Of the Production of the Angels in the Order of Natural Being, a. 1 Whether the angel has a cause of his existence? c.):

It must be affirmed that angels and everything existing, except God, were made by God. God alone is His own existence; while in everything else the essence differs from the existence, as was shown above (Question [3], Article [4]). From this it is clear that God alone exists of His own essence: while all other things have their existence by participation. Now whatever exists by participation is caused by what exists essentially; as everything ignited is caused by fire. Consequently the angels, of necessity, were made by God.

Regarding "God creat[ing] angels out of a river of fire in heaven:" This cannot be because God creates angels out of nothing (ex nihilo), not with matter.

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The belief in guardian angels is not a dogma, but that does not necessarily imply that it's a false belief. Each man has an angel assigned to him, either at baptism or birth (Catholic theologians have different opinions on when the assignment of guardian angels occurs). However, the number of souls God will create before the end of time is not known, since humans are free co-creators in generating a new human being, and Catholicism does not believe in the pre-existence of souls; God creates new souls out of nothing at the very moment of a new human being's existence. Since the number of human souls and angels that can be assigned their guardians are both finite, it seems there might come a time, when human souls outnumber angels, in which God would have to create new guardian angels from nothing.

God creates human souls out of nothing everyday, so it seems does the same for creating new angels.

The short answer is that the Church does not have a definitive teaching on the O.P.’s question: whether God can create “new” angels. However, the perennial philosophy that is the basis for the Church’s theological reflections suggests that, although God has the power to create as many angels as He wishes, from our point of view, the creation of the angels is already “done.” See the fuller answer below.

Man certainly does not have the power to create angels. Angels are, by nature, far superior to man. No creature can create another creature, much less a creature superior to itself.

The Church’s teaching on the angels and their creation

According to the Catholic Church, God is certainly the creator of the angels. However, there are some things to keep in mind.

First of all, the angels are pure spirits; that is, they do no have bodies, as we do, and are free from limitations that a body entails. (See Catechism of the Catholic Church [CCC] 328-330.)

Angels and their relationship to earthly time

On the level of official teaching, the Church does not address the O.P.’s question directly. However, using good theology, based on sound philosophy, one can draw some conclusions.

Time is relative to the one who experiences it

Since angels do not have bodies, as we do, their experience of time is entirely different from us.

Moreover, God is the creator of time; hence He is entirely outside of time. (See CCC 205.) He is also utterly one and simple (see CCC 202). Consequently, on God’s part there is only one act of creation, which identical with His very Essence. (See, for example, Summa theologiae [S.Th] Ia, q. 13, a. 7, 1um and ad 1um [the first objection and the answer to that objection].) That single act produces many effects—namely, us creatures—but the act as such is only one.

Therefore, it is an anthropomorphism to think of God’s creative action as occurring in time. The effects of that action take place in time, but the act as such is eternal: it is, ontologically speaking, God’s very Essence and Being.

What exactly is time?

It is useful, then, to take a moment to consider what time actually is. Aristotle defined time as “the measure of change according to before and after” (Physics IV, 11, 219b1-2) a notion fundamentally taken up by both St. Augustine (see Confessions, XI, cc. 1-28, especially chapters 24-26) and Aquinas (see In IV Phys., lc. 15-23). In other words, creatures experience time to the degree that they experience various types of changes (“movement,” in the classical terminology).

(It is interesting to note that all of our differing ways to measure time all consist in taking a particular kind of change or movement as the basis for measuring other movements. For example, we use the motion of the moon and the Sun as the basis for the length of our days and months; and other regular changes found in nature—such as the oscillations of cesium atoms—to make more precise measurements.)

How different kinds of creatures experience time

As a consequence, different kinds of creatures experience time in different ways, depending on the kinds of changes that they are capable of experiencing. Human beings, like all material creatures, undergo constant interactions with their environment. Hence, they experience time as a continuous movement.

On the other hand, angels, being pure spirits, do not experience a “continuous” time, but, if you will, “discrete” time. It depends on the changes that can take place in them: the decisions of their will, the knowledge infused into them, and so on. Since they are not material creatures, they do not have an “environment” to interact with; and indeed, their entire existence is characterized by a discrete and finite number of actions. (These actions, however, are far more perfect than our own actions, and hence although the number is finite, what they accomplish through those actions is far greater than we ever could in our lifetime. Aquinas makes an extensive treatise on the angels in qq. 50-64 of the Prima Pars of the Summa theologiae. The most relevant questions are 54 and 59, on the angelic intellect and will.)

Angelic time is simply not in the same “continuum” as our own time—it is entirely extrinsic to ours.

That does not prevent angels from interacting with us: but they are looking at our continuous time, as outsiders looking in, so to speak.

From our perspective, God has already “finished” creating the angels

What that means, from our perspective, therefore, is that God already “finished” creating the angels, even “before” He created the material universe. (In reality, “before” and “after” are very misleading here; were are talking about kinds of time that are simply incommensurable.)

Even if God wanted to create more angels—and God is quite capable of creating as many as He wants, even an infinitude, if He wishes—it would seem to us (who exist in our continuous, material time) that He has “already” created them.

That is not a defect on God’s part, as if He were “incapable” of creating angels in earthly time, but simply reflects the difference in nature between angels and men: angels, being pure spirit, are simply not bound by earthly time.

Conclusion

God has the power to create as many angels as He would like: even an infinitude. However, angelic nature is so different from ours (since angels are pure spirits unbound to bodies) that their experience of time is simply incommensurable with ours. Hence, it would be anthropomorphic to say that God creates new angels in our own, earthly time. From our perspective, the creation of the angels is already “done”—but this is not because God cannot create “more” of them, but because our time and their time do not overlap and is of a different nature.

As far as man creating angels, that is impossible, as only God has the power to create anything (in the strict sense—all that we humans can do is rearrange things that have already been created). See CCC 295-301 for an overview of the Church’s teaching on creation.