According to Catholicism is feeling nostalgic a sin? According to a Doctor of Church (e.g., St. Thomas Aquinas), is feeling nostalgic a sin?

I think it is, because saints have advocated detachment from the world, and it is a constant theme of even Christ Himself to detach our hearts from all worldly things and keep our conversation in heaven, as St. Paul reiterated.
In a word, no.
First of all, merely feeling anything cannot possibly be a sin, because sin necessarily entails an act of the will.
The Catechism of the Catholic Church (referencing two great doctors, St. Augustine and St. Thomas Aquinas) defines sin as

an offense against reason, truth, and right conscience; it is failure in genuine love for God and neighbor caused by a perverse attachment to certain goods. It wounds the nature of man and injures human solidarity. It has been defined as “an utterance, a deed, or a desire contrary to the eternal law” [St. Augustine, Contra Faustum 22:PL 42,418; St. Thomas Aquinas, STh I-II,71,6.] (No. 1849, my emphasis).

For an action to be truly an offense, it must entail an act of the will. (Note that “desire” in this definition refers to willful desire, not the feeling of desire.)
Even supposing that the feeling of nostalgia is fostered by an act of the will, there is nothing necessarily wrong with it. It does not offend God or neighbor in any way.
According to dictionary.reference.com, nostalgia can be defined as

a wistful desire to return in thought or in fact to a former time in one's life, to one's home or homeland, or to one's family and friends; a sentimental yearning for the happiness of a former place or time.

There is nothing disordered as such about having a certain sadness for the absense of past happiness—it is (as far as it goes) a normal part of the human condition.
When the Church talks about “detachment from the world,” she means detachment from disordered desires—things that could, if fostered, lead to sinful behavior.
Here is what the Catechism states about detachment:

Jesus enjoins his disciples to prefer him to everything and everyone, and bids them “renounce all that [they have]” for his sake and that of the Gospel. Shortly before his passion he gave them the example of the poor widow of Jerusalem who, out of her poverty, gave all that she had to live on. The precept of detachment from riches is obligatory for entrance into the Kingdom of heaven (No. 2544).
All Christ’s faithful are to “direct their affections rightly, lest they be hindered in their pursuit of perfect charity by the use of worldly things and by an adherence to riches which is contrary to the spirit of evangelical poverty” [Lumen Gentium 42 § 3.] (No. 2545, my emphasis).

Hence, the problem does not lie with having affections (which are good, in and of themselves), but rather in directing them in a disordered way.
In any case, merely having an attachment (which is a kind of feeling), in and of itself, is not a sin; a sin consists in acting in a way that offends God.