Is the humming by the kohanim during birkat kohanim considered a hefsek of their bracha? Please be clear what halachik reasoning allows or disallows this humming.
The Mechaber rules (OC 65:1) that if someone pauses in the middle of a mitzva an amount of time in which he could have performed the entire mitzva (henceforth: a long pause) he does not have to start over with the exception of Tefillah (ie Shmoneh Esrei) where one would have to go back. The Rama rules that for deoraita requirements, one would have to go back if the pause was forced (באונס) and not voluntary. For derabanan requirements he would allow one to not go back, per the Mechaber, and make one always go back for Tefillah, also per the Mechaber. (All the above is as explained in Biur Halacha to OC 65:1 sv Kar'ah.) The amount of time sung by the Kohanim is IMO definitely longer than the time it would take to just say all three pesukim responsively.
So we have three issues:
1) Is this singing considered a forced interruption? The Magen Avraham quotes two opinions if forced means that one was halachically unable to continue (eg. something unclean passed nearby) or if one was practically unable to continue (eg. armed robbers passed nearby). He seems to side with the former definition while our case would seem to be more similar to the latter (the Kohein could go ahead and say the final word loudly while everyone else was singing, I suppose).
2) Does Birkat Kohanim count as Tefillah and would therefore need to be restarted even without a forced long pause for the Rama? So I can't say for sure, but it wouldn't surprise me at all if it were. Consider the full name of the Rambam's section: הלכות תפילה וברכת כוהנים. Birkat Kohanim is said as part of Shmone Esrei, and utilizes the Chazzan and Minyan to do so. The Mishna in Rosh Hashana 4:5 even calls the last Bracha of Shmone Esrei (about peace) "Birkat Kohanim". So the two are closely related. I'm not sure though if that carries over into the laws of Hefsek or not.
3) Does the length of time to say all of Birkat Kohanim include the singing, since that is the custom? The Biur Halacha there mentions this as a possibility but leaves the matter for further study (צ"ע). I don't get the feeling he is particularly convinced by this option.
So if you assume Birkat Kohanim is not Tefillah and you either hold like the Mechaber or hold like the opinion that this type of case is not considered forced, then even if you hold the time spent singing does not combine to the action's length you would not have to go back and restart Birkat Kohanim for singing too long.
All this is at least on a bedieved level. The Aruch haShulchan (OC 65:5) writes that for sure in all cases one should not make such a long break lechatchilah and it is only permitted to do so for great need.
Does Birkat Kohanim count as great need, ie can a Kohen go up if he knows singing will take place? Well, it is a mitzva deoraita, but on the other hand, if a he were to leave the Minyan before the start of the 3rd to last blessing of Shmone Esrei (Retzei...) he would not be obligated in this mitzva, so no loss. CYLOR for a final ruling.
Does reciting the prayers regarding dreams count as a great need, ie should a shul abolish the minhag of singing? Well, it is an old minhag, and the dream-prayers date back to at least the Gemara (Brachot 55b), but on the other hand see the Levush OC 128:26 who records the minhag of the Kohanim singing "the ך in וישמרך, the ך in ויחנך and the ל in שלום [because that way they have finished the bracha at hand (groan)]" thereby avoiding extended pauses. CYLOR for a final ruling.
It seems from this answer that making sounds that are not words does not constitute a hefsek in general. This answer indicates that what makes something an interruption is the fact that it distracts one from what one was doing, so something that doesn't change the subject would not be an interruption.
I spoke this out with a Rov who said this is simply the Minhag and says that the Cohanim should m'varech (bless) the am (people) b'ni'imah (melodically). Although a "source" I don't have for this.