1. In every tefilla and every kaddish is a prayer for moshiach and I was always wondering why do we need to daven for Moshiach so much? If the whole purpose of coming onto this world is to learn Torah and do mitzvos, then let it continue this way. People are born, do their tafkid and go to Olom Habo forever. How does Moshiach come into the picture? 2. Also, in Aleinu, we daven that the whole universe should recognize that Hashem is The King. Wont they recognize this when they die even without Moshiach coming? 3. I was also wondering, if after 120, the tzadikim are in Gan Eden being Nehene miziv Hashechina, why should they want to come back in their physical bodies to this Olom Hasheker by Techias Hameisim where they cant get that schar - schar mitzve Bhai alma Leke? Why can’t they continue in Olom Habo infinitely?

1-2. You are correct that the main reason we do mitzvot isn't to bring Mashiach and not even to go to Olam haBa, but rather to be good and Godly people, and make this world a better place (the reward will come, but that's not what should preoccupy a mature, idealistic adult). Most prayers are actually for "the Days/Period (!) of Mashiach", which include the different stages of redemption= that it should be good for the Nation of Israel: the flourishing of the Land of Israel; mass aliya to Israel from all over; reviving the Sanhedrin; rebuilding the Beit HaMikdash, and only at the end, the Mashiach. All of these 5 stages are each mitzvot in unto themselves, which we've always been obligated to do (build up Israel, live in Israel, set up a Sanhedrin, build a Mikdash, and anoint a king from the House of David), but have unfortunately been technically prevented from practical observance for 1900 years. We have been praying that their practical observance will return asap, and that observance is aligned with the redemption. In fact, the Land of Israel is BH already flourishing and most of the Jewish people already live here, yet we still pray even for those stages to be completed and to begin the next phases. Accordingly, as opposed to Christianity, in Judaism the main point of the Days of Mashiach, is not Mr. Mashiach himself (which is just 1, and probably the last of the 5 aforementioned stages and mitzvot), but rather that we ask God to help us rise spiritually and nationally and help us achieve and fulfill those mitzvot and messianic stages asap, speedily in our days. All of this is in this (!) world, and we are not supposed to wait or rely on what God will do in the world to come. This here is the world of free-will, where God decided that we are responsible, and should act Godly, to destine our fate and improve the world asap, as opposed to the next world where God does everything. As the Kotzker Rebbe said: "If every Jew would treat his neighbor as if he (the friend) were Mashiach, it wouldn't matter even if he isn't!" (for that would precisely be the world that we yearn for). This beautiful world is for us and the gentiles, as well, who should learn from our example. 3. The ideal world is to return to the level of Gan Eden in this world in the Revival of the Dead (Techiyat HaMeitim), living with healthy souls in healthy bodies, which is more complete and on a higher level than in the World of the Souls in Heaven, and accordingly, is the final eternal world (this is the mainstream opinion in the sages, so don't let the Rambam's opinion confuse you…).