(a) Why have I only seen this in Sefard siddurim?
It's also in the Siddur compiled by the first Chabad Rebbe. He mainly based his Siddur on the Siddur Ha-Ari from the Ari Za"l, (who was one of the foremost Kabbalists of recent history), so it's possible that it came from there, although I don't have a Siddur Ha-Ari to verify.
But to answer your question, the Chabad Siddur and Nusach Sefard are both Chassidic. The Ba'al Shem Tov, founder of the Chassidic movement, brought Kabbalah to the masses, in a way they can understand, through teaching Chassidus which is the inner dimensions of all 4 parts of Torah, פשט (simple meaning), רמז (hints), דרוש (expounding) and סוד (Kabbalah). Being that the Chassidic movement was more into Kabbalah than regular Jews, the Chassidic Siddurim resembled closer to Kabbalah.
The paragraph that you're asking about is from Zohar, the main Kabbalistic text. So Chasidim, being more into Kabbalah, used it in the Siddur.
(b) Why is this passage only printed for erev Shabbat? It seems like if its recitation is an appropriate option at one time it should be good for any time. Is there something special about erev Shabbat that makes having no Barechu or 'Barechu-like' liturgical piece especially problematic?
This is the first sentence of the paragraph:
וְלוֹמַר בָּרְכוּ אֶת יְיָ הַמְבֹרָךְ, אֶת דַּיְקָא דָא שַׁבָּת דְּמַעֲלֵי שַׁבַּתָּא:
And let us say barechu es Hashem ham'vorach, ....... Shabbos, the eve of Shabbos. (The ... is where I wasn't able to translate, I don't know Aramaic to well.)
So, the verse is only taking Shabbos eve.