Kuf means "monkey," which is one of the primary symbols of impurity, Klippah, which itself begins with a Kuf. The Kuf is shaped like an imperfect Heh (which represents holiness), just like a monkey is an imperfect imitation of a human being. At the times that we behave properly, the Torah states that five (gematria of Heh) of us will chase one hundred (gematria of Kuf).
On the other hand, Kuf can also stand for holiness itself, Kedushah, which also begins with the letter Kuf. We therefore see that the Kuf has potential for both holiness and unholiness, and represents the process of transformation from unholiness to holiness, just as during these days between Passover and Shavuot the Jews went from the 49th level of unholiness to the 49th level of holiness. Rabbi Yitzchak Ginsburgh in his book, The Hebrew Letters, states that the Kuf represents the kabbalistic concept of "Redemption of Fallen Sparks." (p.280) In Kabbalah, redeeming the holy sparks is the very reason for our existence (Tikkun Olam, "fixing the world"), and the rationale behind our exile(s).
The next letter, the Reish, also represents a similar dual concept. It can stand for Rash (poor) or Rosh (head), just as the month of Nissan itself is both the head of all the months and yet a month of humility in which we eat the bread of poverty. Nissan represent Judah, the head of all the tribes, and yet someone who was humbly willing to accept his shortcomings and transform them. Similar to the Kuf, Rav Ginsburgh states that the Reish stands for Avodat HaBerurim (the service of clarification), which is also very much related to the redemption of the sparks mentioned above. Once the Avodat HaBerurim is completed, Mashiach (son of David, from Judah) will come and bring about the ingathering of the exiles and redemption.
Furthermore, the Zohar mentions that two letters Kuf and Reish together also have a poor connotation. They form part of the word Sheker, a lie. Kuf and Reish by themselves spell Kar, coldness, also associated with impurity (Raskin). Kuf and Reish are also the first two letters of the word Keri, a strong form of impurity associated with seminal emission, as well as with Amalek. Yet, when the last letter of the word Keri, the yud (which, like the Heh, stands for G-dliness) is placed in the beginning, in front of the Kuf and the Reish, it forms the word Yakar, which means "dear." Here too, we see that impurity can be transformed into a feeling of dearness and closeness to G-d.
A similar theme can be found in the Perek Shirah verses of the Wind and the Lightning Bolts:
The verses above are clearly related to the ingathering of the exiles. Both verses speak of the "ends of the earth." This is related to Passover, but also to Yom Ha'Atzma'ut. (See here, how theoretically Yom Ha'Atzma'ut could be celebrated as late as the 9th of Iyar, the 24th day of the Omer).
Wind in Hebrew is "Ruach," which also means spirit. It is a word specifically connected to Mashiach, and the Haftorah we read for the last day of Passover. The verse of the wind specifically addresses two kinds of exile, north (Assyria) and south (Egypt), telling the forces of impurity to "give up" and "not withhold," elevating the sparks and transforming them into holiness.
The Lightning Bolts also bring to mind the miracles of Egypt and the giving of the Torah at Sinai (marked by both thunder and lightning). The verse also speaks of the Lightning Bolts making "vapors" ascend, which seems very much parallel to the concept of elevating the fallen sparks back to their source. In fact, Rav Ginsburgh mentions "vapor" as an aspect of elevating fallen sparks, related to both the Reish itself and the form of the Reish within the Kuf itself (made of a Reish and Zayin). Interestingly, the verse of the Lightning Bolts also mentions the wind.
The Temple guard for these 22 days is connected to the priestly family of Seorim. Seorim means sheaves of barley, which is exactly the material used for the Omer offering. The Omer is referred to in the Torah as 'Minchat Seorim," an offering of barley. This is connected to the Counting of the Omer done at this time of year.