Dear Rav Mordechai,
The Gemara cites Rav Yitzchak as saying that the 7/5 division of the showbread was "schar hagafat delatot". - But I vaguely remembered that "hagafat delatot" was a Levi job, not a Cohen job!
So, I did a computer check [not-exhaustive], and found my memory wasn't too bad: Rashi on Brachot 63B DH Yinagen Kinor cites Erchin 12B, where Rashi comments how some Levi'im were singers and some were doormen (for lack of a better translation). Tosfot in Erchin cites a source in Divrei HaYamim which indicates that the different jobs were assigned to different families.
I suppose one could say that Rashi in Sukka 56B says Mishmar, not Mishmar Cohanim, and parallel to the Cohanim there were Mishmarim and Ma'amadim of Levi'im and Yisroel'im, but that wouldn't explain why Cohanim would get schar for the Levi'im's work. (I don't remember learning that Levi'im and Yisroel'im swapped duties when Cohanim did, and to me at least it doesn't necessarily follow that they would have to).
RASHI in Yoma (18a) mentions that it was the responsibility of the Kohanim to open the doors to the "Azarah, Chayil, Heichal, and Lishkos." When the Gemara in Erchin (11b, not 12b) mentions that the Leviyim had the job of opening the doors, it may refer to other doors, such as the doors into Har ha'Bayis.
All the best,
(a) Thanks, Rav Mordechai. The Svora fits, though it leaves Rashi without a source (at least that I could find) on which to base himself. All the other incidences of discussion of Shoarim (that I found) related to Levi'im. Of them, one Metzudat David (Divrei HaYamim I 9:18) DOES qualify the job to "Har HaBayit SheHi Machane Levia", but he can't be a source for Rashi, and he also doesn't cite a source [and since the Chayil was Machane Levia, he doesn't totally concur with Rashi Yoma 18A]. In some incidences, I found even Rashi seeming to indicate otherwise, such as in Divrei Hayamim 2 31:2 "HaKohanim LiSharet, VeHaLevi'im Lehiot Meshorarim VeShoarim". (I feel uncomfortable resorting to using these computer Torah databases - I'm afraid they can easily mislead that one has done a thorough search, or knows all the sources). Finally, the only Rambam I found on the subject (Klei Mikdash 3:2) makes no distinction and seems to consider only Levi Shoarim.
(b) That left me wondering in what regard the job was held. The Gemara (somewhere) says that Levi'im who couldn't sing would become sho'arim, possibly indicating that it ranked second fiddle (rightly or wrongly -and almost a pun) to being a member of the choir. Could it be that since it was neither part of the Avodah, nor presumably needed in the Mishkan, it was considered a post-Mishkan "extra job" that Kohanim had to do and that couldn't be delegated to Levi'im only because Levi'im weren't allowed in those places [doesn't explain the Chayil, though], and thus the Kohanim got a 'bonus', as it were, in the form of a showbread advance? A 'procedural bonus' that would seem to me to be a lot easier to legislate than some 'substantive bonus'.
(a) Rashi indeed has clear sources. The Mishnah states in great detail that the Kohanim were the ones who took the keys and opened the Heichal doors (Tamid 30b) and that the "Ziknei Beis Av" of the Kohanim were the ones who guarded the keys to the doors of the Azarah (ibid. 25b).
All the sources that you cited which defined Leviyim as Sho'arim, probably indicate simply that the Kohanim were not designated for specific tasks. Any Kohen can either open the doors, offer Korbanos, etc. (as the Mishnayos, ibid., seem to show). The Leviyim, on the other hand, were required to stick to their jobs, "Meshorer she'Shi'er b'Misah" (Erchin 11b). Therefore, only the Leviyim included people who were specifically designated as "Sho'arim."
Rav Yehudah Landy further pointed out to me that the Pasuk also mentions specifically that the Kohanim were appointed to opening doors. In Malachi 1:10, the prophet reprimands the Kohanim for "not wanting to close the doors for free." As the Toras Kohanim explains (Tzav, Parshah 11:10), Hash-m was reprimanding them for demanding compensation even for the type of activity that a person does for his friend for free.
As for your question about the Chayil, I don't know what Rashi's source is that the Kohanim were in charge.
(b) I don't remember seeing that closing the gates was relegated only to Leviyim who could not sing. (It should be given to strong Leviyim, regardless of their voice.)
Thanks very much.
The only source I found to support my (too often incomplete) memory is Ramban Clei Mikdash 3:8 Seifa.
Baruch - I couldn't find a source for your statement that closing the gates was a "second class" job, given only to Leviyim that could not sing. The Rambam (see Insights to Chulin 22) simply says that the Leviyim who were too old to sing were allowed to work with the gates. Leviyim who were young enough to sing may alsohave been given the job of closing the gates.
Be well, Mordecai