Seder Haboker

Mah Tovu is said by every individual when he enters Shul.
Adon Olom is the beginning of Shacharis, said by the Chazzan Tachton (second Chazzan) by the Omud. next come Al Netilat Yodayim, Asher Yotzar, and Elokai Neshomo, and then Birkos Asher nosan Lasechvi till Gomel Chassodim Tovim and Yehi Rotzon.
After that comes Ribon Kol Ho'olomim till Omar Hashem, Birkos Hatoroh, Parshas Hatomid, The Posuk Veshochat Oso, (On Shabbos Uvyom Hashabbos and on Rosh Chodesh Uvroshei Chodsheichem), Eizehu Mekomon and Bishlosh Esre Midos Hatoroh Nidroshes, and Yehi Rotzon Sheyiboneh Beis Hamikdosh. Next come Birkos Tallis and Tefillin, and then Boruch She;omar and Psukei Dezimroh.

According to Minhag Ashkenaz, Psukim said Derech Tefilloh do not require Birkos Hatoroh before them. Minhag Polin was more strict regarding Psukim before Birkos Hashachar, therefore Birkos Hatoroh was said after Asher Yotzar (and later moved to after Elokai Neshomo), to make it would be before the Posuk at the end of Ribon Kol Ho'olomim "בעת ההיא אביא אתכם ובעת קבצי אתכם" etc., but even according to Minhag Polin Mah Tovu was said before Birkos Hatoroh.

Michael

All Brochos are said in shul, this way one can answer Omen on them. See Shulchan Oruch Siman ו seif ב who brings this Shito (and mentions that the Sephardim have a different Minhag).
The R'mo in Siman מו Seif ט says that Minhag Ashkenaz is that Psukim said דרך תחנונים can be said before Birkos Hatoroh.

Michael

The reason the Brochos are said out loud is so the Tzibbur can answer Omen. The Tzibbur usually will say the Brochos before the Chazzan starts, and this way one can hear all of the Brochos and answer them properly.

Michael FRBSH