Tashlich - Yechaveh Daat 1:56

In this teshuva, Rav Ovadiah Yosef begins by explaining the source for the universally accepted practice of tashlich. Citing the Maharil, he notes that it is based on the Midrash which explains that when Avraham and Yitzchak went to do the akeidah, the Satan tried to stop them by turning into a river and blocking their way. Avraham reacted by calling out to Hashem and saying that if they were to drown in trying to fulfill God's command, then who would be left to publicize the name of Hashem in the world!? Thus, as a way of recalling the merit of this most incredible of religious acts (which Chazal believe took place on Rosh Hashana), we say this tefilla by the banks of a river or stream.

A second reason given is that since kings were anointed by the banks of a river, we as it were "anoint" Hashem on this day that proclaims His dominion over the world.

In terms of whether or not tashlich should be said on Shabbat, Rav Yosef acknowledges that there may be some esoteric reasons for not doing so, but he sides with the view that the main reason to not do so is that people will carry their machzorim to tashlich and thus will violate the prohibition of carrying on Shabbat. As such, if the body of water being used is within the eruv, it would seem permissible to say tashlich even on Shabbat.

Blowing Shofar during Shemoneh Esrei - Igrot Moshe Orach Chayim 1:173

There is a practice to blow shofar not only before שמונה עשרה and not only during חזרת הש"ץ, but also during the silent שמונה עשרה. In this practice, people reach a certain point in their davening, then stop and wait for the shofar to be blown, then resume their silent prayer. Rav Dovid Feinstein asked his father what the law is regarding those individuals who daven slower and thus are still in the middle of a bracha when the shofar is blown. Is their pause in order to listen to the shofar and thus fulfill the mitzva considered to be an interruption in their davening?

Rav Moshe answered that there is no problem of הפסק, interruption, for two main reasons. First, even though we have a principle of שומע כעונה, under which a person is considered to have said a bracha merely by listening to another recite it, in this case the person is not considered to have said anything or even to have blown the shofar. He is merely listening to the sound and thereby fulfilling his obligation. Mere listening does not rise to the level of interruption.

Second, Rav Moshe reasons that the entire silent שמונה עשרה is considered to be an appropriate place for the blowing of the shofar, and thus wherever in davening one is up to when the shofar is blown, that is a suitable spot for him to hear it, and thus there is no הפסק issue.

Paying chazzanim - Yechaveh Daat 1:53

Is a community allowed to pay the individual who serves as the בעל תפילה or בעל תוקע on Rosh Hashana? The question is based on the gemara in בבא מציעא which states that even an action that is completely permitted on Shabbat cannot be performed for compensation unless the compensation also encompasses a time which is not on Shabbat (or Yom Tov). With the individuals in question, it would seem that their entire task is executed on Yom Tov and thus would be in violation of this rule.

Rav Ovadiah Yosef deals with this issue by noting the long history of Rishonim and Acharonim who permitted such practices, on the basic grounds that this prohibition does not apply when the action in question is for the purpose of performing a mitzva. He notes that even the Shulchan Aruch only says that the compensated individual will not see any bracha from that money, yet he refrains from saying that such an action is forbidden. Many others, including the מהר"ם מרוטנברג and Rav Shlomo Kluger, permitted paying individuals to lead the davening on Shabbat and Yom Tov without any qualifications, and thus Rav Yosef feels that we can comfortably be lenient in this regard.