Shin literally means "tooth," the strongest bone in the human body. It is also one of the "mother" letters of the Hebrew alphabet, and is connected to Esh, fire. According to Sefer Yetzirah, it represents the head.
The Shin has different ways of being written (sometimes with 4 arms instead of 3), and also has different sounds (sometimes Sh, sometimes S, and sometimes silent). The Shin's numerical value is 300, consistent with its three-armed shape. This fiery letter sometimes has a negative trait. Along with the two letters of the previous cycle, it forms the word Sheker, a lie.
These are fiery times in the Jewish calendar, particularly Lag Ba'Omer (also very much connected to the number 3, as it is the 33rd day of the Omer: Lamed is 30 and Gimmel is 3. Yom Yerushalayim is also connected to fire. Of the 4 holy cities, Jerusalem is fire. "It was destroyed by fire and will be rebuilt by fire."
The next and final letter in the Aleph-Bet is the Tav. Is in many ways the opposite of the Shin. It represents Tmimut (wholesomeness), perfection, stability. Along with the "mother" letters Alef (air) and Mem (water), it spells out the word Emet, truth. As the final letter, it represents completion. It also represents death, Mavet, which ends with the letter Tav. This cycle is also about completing the Counting of the Omer, which are called Shevah Shabatot Temimot (seven wholesome, complete weeks). Also, as explained in the previous cycle, it is during the counting of the Omer that we transform unholiness into holiness, falsehood (Sheker) into truth (Emet), and, ultimately, death into life. As we encamp at Mt. Sinai on Rosh Chodesh Sivan, we are settled, stable, and in harmony, prepared to receive the Torah.
A similar theme can be found in the Perek Shirah verses of the Dew and the Rain:
Dew represents renewal. It is connected to resurrection of the dead. There is also something called the Tal of Torah. The teachings of Rabbi Shimon Bar Yochai revived us. We blossomed like a rose... which are the opening words of the Zohar.
The Dew also connotes a "surprise" redemption, which is independent of our merits. Such was the conquest of Jerusalem and the Six-Day war, in which Israel "spread forth his roots as the Lebanon. The "Lebanon" is a known reference to the Temple and Jerusalem.
The additional text also seems related to Lag Ba'omer and Yom Yerushalayim, as it speaks about being aroused from the hidden (North, Tzafon, can also mean Tzafun, hidden), like the hidden knowledge of Kabbalah revealed by Rabbi Shimon Bar Yochai. It also speaks of spreading out to the north, south, and to Hashem's garden, Jerusalem.
Rain is usually conditional on our behavior. It is in this way more connected to the attribute of judgement and truth, like the Tav. However, there are times, like on Yom Yerushalayim, when even rain can fall simply out of Hashem's generosity, as well as out of the need to protect His heritage. As we say in the opening lines of the Shmoneh Esreh Prayer... "Elohei Avraham, Elohei Yitzchak, veElohei Ya'akov... uMevi Goel liVnei Bneihem, L'ma'an Shemoh Be'Ahavah" (the G-d of Abraham, the G-d of Isaac and the G-d of Jacob... and Who brings redemption to their children's children, for the sake of His Name, in love). The song of the Rain is also about us feeling strengthened and firmly established, like the encampment by Mt. Sinai.
These last two elements also mark the end of the second chapter of Perek Shirah, and the completion of all the songs of the elements of the mineral kingdom. The following chapters in Perek Shirah relate to trees and to animals.
The Temple guard for these 22 days is connected to the priestly family of Miyamin. This name is spelled the same and appears to be a reference to the first word in a very powerful verse in the song of Haazinu: "Miyeminoh Esh-Dat Lamo" (At His right hand was a fiery law unto them). This hints to many of the ideas mentioned above. This is a time in the Jewish calendar that is very much connected to fire and to Torah, as it includes Lag Ba'omer, Yom Yerushalayim, and Rosh Chodesh Sivan, leading up to Shavuoth. As mentioned above, the days above also contain an aspect of G-d's mercy, symbolized by His right hand.