To add to the other answers - here's a Halacha that has been overlooked:
וְלֹא יְנַגְּנוּ הַכֹּהֲנִים אֶלָּא נִגּוּן הַמְיֻחָד, מִפְּנֵי טֵרוּף הַדָּעַת.
The Cohanim should only use the standard tune(s) so as not to get confused.
IOW: It's improper to start introducing new tunes for Birkat Cohanim; each shul should keep to its traditional ones so as not to get the Cohanim confused.
Source: Kitzur Shulchan Aruch 100:12
שְּׁלִיחַ הַצִבּוּר מַקְרֶא אוֹתָם בִּרְכַּת כֹּהֲנִים מִלָּה בְּמִלָּה, וְהֵם אוֹמְרִים אַחֲרָיו כָּל מִלָּה, עַד שֶׁמְסַיְּמִים פָּסוּק רִאשׁוֹן, וְעוֹנִין הַצִּבּוּר אָמֵן, וְכֵן אַחַר הַפָּסוּק הַשֵּׁנִי וְאַחַר הַפָּסוּק הַשְּׁלִישִׁי. לֹא יִקְרָא שְּׁלִיחַ הַצִבּוּר בְּעַל פֶּה, אֶלָּא מִתּוֹךְ הַסִּדּוּר, שֶׁלֹּא יִתְבַּלְבֵּל. וְיָכוֹל לוֹמַר גַּם הוּא הָאֲמֵנִים שֶׁלְּאַחַר הַפְּסוּקִים וְלֹא הֲוֵי הֶפְסֵק, שֶׁזֶּהוּ צֹרֶךְ תְּפִלָּה. אֵלּוּ תֵבוֹת שֶׁהַכֹּהֲנִים הוֹפְכִים בָּהֶן לַדָּרוֹם וְלַצָּפוֹן, יְבָרֶכְךָ, וְיִשְׁמְרֶךָ, אֵלֶיךָ, וִחֻנֶּךָ, אֵלֶיךָ, לְךָ מִשּׁוּם דְּתֵבוֹת אֵלּו הֵן לְנֹכַח, לָכֵן הוֹפְכִין אֶת עַצְמָם גַּם לַצְּדָדִין, כְּדֵי לְבָרֵךְ אֶת כֻּלָּם. וְכֵן הוֹפְכִין גַּם בְּתֵבַת שָׁלוֹם, לְפִי שֶׁהִיא סִיּוּם הַבְּרָכוֹת. וּבְשָׁעָה שֶׁמַּאֲרִיכִין בְּנִגּוּן שֶׁל הַתֵּבוֹת שֶׁבְּסוֹף הַפְּסוּקִים, דְּהַיְנוּ, וְיִשְׁמֶרְךָ, וִיחֻנֶּךָ, שָׁלוֹם, אוֹמְרִים הַצִּבּוּר רִבּוֹנוֹ שֶׁל עוֹלָם וְכוּ'. וְהַמַּקְרֶא, אֲפִלּוּ אֵינוֹ שְּׁלִיחַ הַצִבּוּר, לֹא יֹאמַר, רִבּוֹנוֹ שֶׁל עוֹלָם, מִפְּנֵי הַטֵּרוּף. וּמִכָּל שֶׁכֵּן שֶׁאִם הוּא שְּׁלִיחַ הַצִבּוּר, שֶׁלֹּא יֹאמְרוֹ מִפְּנֵי הֶפְסֵק בַּתְּפִלָּה. וְלֹא יְנַגְּנוּ הַכֹּהֲנִים אֶלָּא נִגּוּן הַמְיֻחָד, מִפְּנֵי טֵרוּף הַדָּעַת.
The Chazzan reads to them the Birkas Kohanim word for word, and they repeat each word after him until the conclusion of the first verse, whereupon the congregation responds Amein. [In this manner, the Chazzan reads the second and third verses as well.] After the second verse and after the third verse, [the congregation also responds Amein.] The Chazzan should not recite (these verses) from memory but should read them from the Siddur in order not to get confused. The Chazzan, too, may respond Amein after these verses. It is not considered an interruption because it is necessary for the prayer. When reciting the following words, the kohanim turn (partially) toward the South and then to the North: Yevarechecha, Veyishmerecha, Eilecha ["May He bless you" "and safeguard you," "to you"] Vichuneka, Eilecha, and Lecha, ["and be gracious to you" to you" "you"] because these words are in the second person. therefore, the kohanim turn also toward the sides, in order to bless the entire congregation. They also turn thus when saying the word Ahalom, because it is the conclusion of the blessings. While the kohanim prolong the slow chant before saying the concluding words of each verse, which are Ve'yishmerecha, Vichuneka and Shalom, the congregation recites (the prayer) Ribono Shel Olam etc. "Master of the Universe …" The one who prompts the Kohanim, even though he is not the Chazzan, should not recite Ribono Shel Olam, in order not to get confused. Certainly the Chazzan should not recite it because it would be considered an interruption of the Amidah. The Kohanim should chant only the customary melody in order to avoid confusion.