I don't really have too many he'oros on Rosh Hashana since for the most part I was busy with my children while my wife took care of the baby and rested. But I figured I had to post something.
1) I bought an interesting sefer this year called Moadim L'Simcha that was put out by the editors of Otzar HaPoskim. It has a collection of essays on the various minhagim of the Yomim Tovim (selichos, tashlich, kapporos, hakafos etc). One of the articles was on the tekios d'm'umad and tekios d'm'yushav. Here are a few interesting points
a) There is a machlokes rishonim exactly when this taknah was developed. The Ramban understands it started with the amoraim while other rishonim hold it started with the tanaim.
b) The reason given for having 2 sets of tekios is "l'arveiv hasatan"-confuse the Satan. The Ramban explains that the point of confusing the Satan is so that he should not be m'katreig on our tefillos. In other words, we blow before musaf so that when it comes time for musaf the Satan will have already been m'katreig on us and he won't be around to stop HKBH from hearing our tefillos.
c) Many Rishonim (the Ramban and others) hold that there are really 2 issues going on. One is the mitzva d'oreisa to hear the shofar and the other is a takana d'rabanan of blowing shofar in the middle of davening but lav davka 30 kolos. This takana is not limited to Rosh Hashana but also applies to a ta'anis tzibbur as well. Originally, once Chazal set up this takana of blowing shofar in the middle of davening, they decided that we should blow 30 kolos to be yotzei the d'oreisa as well. These tekios are known as tekios d'm'umad. At a later point in time, Chazal also instituted tekios d'm'yushav in order to confuse the Satan.
For a more lomdishe approach to this sugyah take a look at my friend the Divrei Chaim.