Misuse, turning potential prosecution into defense. This is why a kohen who committed idolatry (an internal sin) can repent and perform blessings, but one who murdered (with hands as a tangible reminder) cannot use those hands for blessing. Speech sins fall into the former category, allowing the mouth to be redeemed through sincere repentance on this day of judgment.
Additionally, the shofar itself serves as a complementary form of "prayer" that bypasses the mouth entirely, emanating from the heart to provide an untainted channel of supplication, further mitigating any concerns.
See discussions in Rosh Hashanah 26a, Tosafos on Yevamos 7b, and related commentaries like the Be'er Heitev.
v) First of all, the chet ha'egel was a terrible cheit that we were guilty of. No one has yet been found guilty for sinning with their mouth - they also perform many mitzvos with their mouth.
Second, a shofar needs a shiur. If it was used for avoda zarah we say ketusi michtas shiurei and you can't be yotzei with it. But things you said, even if chas shalom for an aveirah, a dibur is not always a maaseh, it has an ephemeral aspect to it. Doing an aveirah with the mouth doesn't "poison" the mouth to disqualify it from being able to daven with it.
Finally, hakol kol Yaakov, and Yisrael af al pi shechata Yu, Hashem gives us the power of teshuva, of which tefillah is an integral part. We say viduy on Yom Kippur for sins of the mouth - eating improper things, saying improper things and lashon hara, etc. But that does not disqualify us from using our mouths for tefillah.
(U’teshuva utefillah utzedaka maavirin es roa hagezeira - if you made your money by stealing, you probably still have some kiyum of tzedokah, but subject to Sanhedrin 6 botzeia berech, ni'etz Hashem.) (R’ Ben Tzion Lazovsky)
vi) Your question bothered me for many years. Especially with the Gemara Menachos (44a): אותן מצעות שהציעה לו באיסור הציעה לו בהיתר זה מתן שכרו בעה"ז ולעה"ב איני יודע כמה. Baalei Teshuva are taught to use the koichos they used for aveirah now for kedusha...for example, instead of dancing in forbidden places, now dance even harder at a chasunah, etc. Ah, but what about this rule? They shouldn't dance at all!
The answer that I found b'siata D'Shamaya is that the rule of ein kateigor naaseh saneigor is only said regarding lifnei v'lifnim. And the shofar has a din of lifnei v'niflim (Yesh omek b'zeh). The Gur Aryeh and the Mizrachi discuss this when Aharon HaKohen paused when performing the service in the Mishkan for the first time. (Ayen sham) (R’ Sivan Simnegar)
vii) When you daven, you are mekayem the pasuk הרחב פיך ואמלאהו. And Hashem is an אש אוכלה. So davening is a hagolah for the mouth which is metaher. (R’ Elliot Liberman)
