Rosh Hashana 34b - Blowing or Blessing

MAZAL TOV ON MASECHES ROSH HASHANA!!!

If faced with a choice of one or the other, one should choose a place where they may blow shofar, rather than a place where they definitely have a shliach tzibur to daven, bec. shofar is d'eoraysa (gemara). The Sefas Emes asks that according to the Rambam that a safek d'oraysa is only derabonon l'chumrah, it comes out that either way it is only derabonon. He is mechadesh that even though the chiyuv is the same one way or the other, one is still required to choose the side where you may walk away fulfilling a mitzvah d'oraysa rather than a mitzvah derabonon.

A second point: The Nishmas Adam (68:3) proves from this gemara that one is not responsible to travel to another city to perform a mitzvah since the city where they have a shliach tzibbur are not traveling to hear shofar in the other city. Based on this, why would the individual have a chiyuv to travel to the city where they are blowing, he should also not be required to travel for a mitzvah!
Perhaps we can slightly modify the sefas emes's peshat and say that on the level of chiyuv there is no difference between the 2 cities since the chiyuv to go to either one is only d'rabonon. Nevertheless, the chiyuv d'rabonon requires this person to travel to one of them, therefore the gemara says there is an advantage in choosing a city where he may fulfill a d'oraysa mitzvah. But, the people of the city where they are blowing do not have chiyuv to travel because on the level of chiyuv, it is derabonon either way (at least acc. to Rambam).

Rosh Hashana 32b - Aseh Docheh Lo Ta'aseh

The mitzva of shofar doesn't push off chilul yom tov because yom tov is a positive and negative commandment. The Ran points out that the gemara does not have to say that yom tov is a positive and negative, there is another technical issue in that the aseh and lo ta'aseh does not happen at the same time and therefore yom tov cannot be pushed off for the sake of shofar. Furthermore the Ran points out that the gemara is making an assumption that chazal made chizuk l'divreihem to be like a d'oraysa.
Although the Ran makes these 2 points independently of one another, to me it seems that the second point somewhat answers the first. The reason why the gemara picks the aspect of yom tov being positive and negative, is that it wants to explain that yom tov is very severe and that is why even a rabbinic violation of yom tov is not transgressed for the sake of a mitzvah of shofar. If the gemara would have picked the technical issue of "b'idna", that is just a technicality in aseh docheh lo t'aseh and would not justify in any way why an issur derabonon cannot be done.
Btw, the magen avrohom (also korban nesanel) answer the question and create a yesod that you don't need b'idna for an aseh to be motzi a rabbim (see chayei adam 68).

Rosh Hashana 29a - Half Slave - Half Free

Todays minchas chinuch-style gemara says that a chatzi eved/ben-chorin cannot blow for himself bec. the tzad avdus will be motzi the tzad cheirus. I didn't get a chance to look around much but it seems pashut that this only applies to the mitzvah of shofar. The mitzvah of shofar is (acc. to most) a mitzvah to hear, not to blow (or at least primarily a mitzvah to hear like by the person coming out of the pit while blowing 28a, see minchas chinuch). Therefore, when someone blows, he is not being yotzei from blowing but rather his own blowing is motzi himself through hearing the sound. The problem is that a chatzi eved will be blowing for the chatzi ben chorin that is listening. But by other mitzvos such as tefillin, when he puts tefillin on his head, the entire chatzi ben chorin is wearing tefillin and yotzei. Similarly if he makes kiddush which is a mitzvah to say (not to hear), he can be yotzei bec. he the chatzi ben chorin is saying the kiddush.

The question is what about megillah. I would say that since the mitzvah of megillah is to read (at least for men), therefore the chatzi eved can read for himself. The tzad cheirus is yotzi through the reading even without hearing. However, the rema (689) writes that some say he cannot read from himself. His source is the beis yosef who says "the tur isn't concerned to point out that a chatzi eved/ben-chorin can't read for himself bec. he relies on what he writes by shofar". If not for the Rema and Beis Yosef, it seems more mistaver that the Tur is very meduyak and that the problem only applies by shofar and not by megillah. Perhaps the Rema is being choshesh that the megillah must also be heard not just read (see m.b. 689:5 by a deaf who is not mute).