2. The Rambam writes that if you blow tekiyos from a stolen shofar you are yotzei because “ain b’kol din gezel” – you can steal a physical instrument, but you can’t steal sound. Ra’avad disagrees and quotes a Yerushalmi that there is no problem is mitzvah haba'ah b'aveira because of a gzeiras hakasuv: “Yom teru’ah” – m’kol makom.
The Rambam is predicted on two assumptions: 1) mitzvah haba’ah b’aveira is a psul in the chefza shel mitzvah; 2) the cheftza shel mitzvah when it comes to shofar is the sound that emanates from it.
You could read the Ra’avad as rejecting the first assumption. Even if only the kol is the cheftza shel mitzvah, the bottom line is that had the shofar not been stolen, there could be no kol. It’s not the object that is the sole focus of mitzvah haba’ah b’aveirah – it’s the process of enabling the mitzvah performance. So you need a gezeiras hakasuv to take off the psul.
But you can also read the Ra’avad as rejecting only the second assumption. Mitzvah haba’ah b’aveira is a psul in the chefzta shel mitzvah – but the Ra'avad is arguing that the cheftza shel mitzvah is not the kol shofar, it’s the day itself! Blowing is just a means to create a chalos of the day being a “Yom teruah,” not an end in its own right. (See R’ Baruch Povarski’s Bad Kodesh #46 for a similar approach.)
3. There is an interesting machlokes haposkim among the sefardi poskim regarding what to do if one forgets and says “melech oheiv tzedaka u’mishpat” instead of “halemech hamishpat” during the aseres y’mei teshuvah. The Mechaber paskens that one must repeat shmoneh esrei, similar to forgetting “hamelech hakadosh.” The Rama disagrees and holds that since you also mention “melech” in the normal chasimah, that is sufficient. Even though sefardim usually follow the Beis Yosef, here the Ben Ish Chai paskens like the Rama based on the rule of safeik brachos l'hakeil. When in doubt (and the Rama's view is sufficient to create at least a doubt as to what the final rule shuld be) one does not repeat a bracha.
One of the themes that runs through R’ Ovadya’s teshuvos is his staunch defense of Maran, the Beit Yosef, and this case is no exception. Let me say off the bat that R' Ovadya does not deny that in principle one can rule against Maran based on the rule of safeik brachos l'hakeil when there are other dissenting views. However, this case is different. R' Ovadya starts with the premis that if one omits or errs in a bracha of tefilah, the entire tefilah is invalid. From here he creatively argues that if one has any doubt whether one is yotzei with the chasimah of "melech oheiv tzedaka u'mishpat," by simply continuing with the rest of shmoneh esrei one is risking reciting all the remaining brachos in vain as the entire tefilah is worthless if the chasimah is wrong. The rule of safeik brachos l'hakeil must be weighed against the potential competing risk of bracha l'vatalah on all the remaining brachos. In effect, the rule of safeik brachos l'hakeil does not apply to tefilah.
A full discussion of all the proofs and counterproofs is beyond what I can summarize here (see the two teshuvos in Yabia Omer vol 2 and in shorter form Yechaveh Da'at vol 1). I will mention one question as it relates to inyana d'yoma. The Mishna in Rosh HaShana (34) writes that brachos and tekiyos on a ta'anis are not me'akeiv, but are me'akeiv on Rosh haShana. You need to say all of malchiyos, zichronos, and shofaros to get credit -- if you only know one of the three, it doesn't do you any good. Same with tekiyos: you need to blow tekiya, teru'ah, tekiya -- if you only know how to blow one and not the other, it does you no good. On a ta'anis, however, even if you only know one of the six additional brachos that are added to the shmoneh esrei, you can say that one addition and omit the rest. The MG"A (siman 593) extrapolates from here that the same is true of any brachos of shmoneh esrei except for those of Rosh haShana, i.e. if you know only some of the brachos of shmoneh esrei and not others, say what you know -- covering everything is not m'akeiv. Clearly this MG"A flies in the face of R' Ovadya's assumption that missing a single bracha renders the entire shmoneh esrei and all subsequent brachos to be brachos l'vatalah. Two possible defenses: 1) perhaps the MG"A's extrapolation from the additional brachos added to tefilah on ta'anis to the core 18 brachos is incorrect; 2) even if the MG"A is correct, perhaps he merely meant that there is some value as a kiyum of "rachamei" in reciting what you know, but even he would not consider that recitation to be a fulfillment of the requirement of tefilah. Be that as it may, there is much more to the discussion.