The order of the requests is explained as follows:
Bracha #5, requesting repentance (teshuva) follows #4, a prayer for under- standing (bina) because we learn from the words of our prophets that understanding precedes repentance. This emphasizes the fact that ideal repentance is not merely an emotion, a subjective feeling of being born again. On the contrary: the basis for repentance is that a reasoned recog- nition is required of how one strayed and why the way of the Torah is proper. (Shulchan Arukh OC 115 states that the reason is that prayer is impossible without understanding; this teaches us that prayer too must engage our reason and not just our emotion.)
The prayer for healing (refua, bracha #8) comes after the request for forgiveness (selicha, bracha #6) to teach the healing power of G-d's forgiveness.
The material blessings of abundant produce which we request in the blessing of the years (mevarech hashanim, bracha #9) is the harbinger of the ingathering of the exiles (mekabetz nidchei amo Yisrael, #10). The Torah tells us that the land of Israel will be desolate during the time of our exile - "And I will make the land desolate" (Vayikra 26:32). The Ramban on Vayikra 26:17 explains that this is not a curse, but on the contrary, a blessing: "For our land does not accept our enemies. And this is also a great proof and promise to us. You will not find in any settled area a good and broad land, which was settled in the past, which is so desolate, for since we left it has not accepted any nation or language; all of them seek to settle it, yet they are powerless to do so". When the land of Israel starts to flourish economically, this is a sign that it is preparing for the return of its people.
Once the Jewish people are gathered in our land, then the judges can pass judgment on the wicked (tzedaka umishpat, bracha #11). This will bring an end to destructive heresy (lamalshinim, #12). This order testifies to the relationship between social justice and religious belief. When people see with their own eyes that a system of justice based on Torah values is able to create a righteous commonwealth, then they will naturally be drawn to believe in it. Conversely, it is inappropriate to expect that we will be able to root out heresy if our tradition and values have not been successful in building a model society.
Once belief in G-d and His Torah are properly established, then the righteous - including the righteous converts - will receive the honor due them (latzadikim, bracha #13).
This will take place in the rebuilt Yerushalayim (#14), and once Yerushalayim is restored, then the Messiah - the Davidic King - will rule (et tzemach David, #15). As we have explained before, the inner message of many laws is that sovereignty in a Torah state is not just a practical convenience; it must be based on devotion to G-d. This is symbolized by the requirement for the seat of kingship to be in the city of holiness.
Once the Jewish commonwealth is restored, the true power of prayer will return to us (shomea tefila, #16). This is the principle at work in the juxtaposition of redemption and prayer. The definitive expressiveness of prayer is dependent on individual and national freedom.
When our power of expression is restored, we will be able to restore the Temple service (r'tzei, bracha #17), enabling us to properly convey our gratitude to HaShem (modim, #18). This indicates that the Temple service is not meant to be some mechanical fulfillment. Each aspect of the sacrificial order has a special inner meaning and message, and we will be able to carry out this service only when we have regained our expressive ability on a national level.
Indeed, there is a unique connection between prayer and sacrifice. The first words of the first verse, describing the first sacrifice to be brought in the Mishkan, read "And this is the thing [vezeh hadavar] which you shall do to them, to sanctify them to serve Me as Kohanim" (Shemot 29:1). "Davar", which we translated as "thing", also means "word". The Midrash Rabba on this verse indicates that this hints at words of Torah or prayer, which HaShem accepts whenever bringing a Korban is not mandated or not possible.
The Priestly Blessing (recited after modim) will be restored in full once the Temple is built, and it is a blessing of peace (sim shalom, #19).