1) Why are the gloves of Yissochor not treated as an issue of Yitur B'gadim rather than chatzitza? Is chatzitza analytically prior in order of p'sulim?
Excellent question! If it was a Yitur Begadim, it should certainly have mentioned it. In fact, Tosfos (Pesachim 57a, Sukah 37a) explains that such a Chatzitzah may not even disqualify the Avodah, while Yitur Begadim certainly will.
I noticed that the Pnei Yehoshua (Pesachim 57a) and the Be'er Sheva (Kerisus 28b) argue with Rashi, who maintains that Yisachar was punished for serving with a Chatzitzah, and explain that he was punished for serving with Yitur Begadim.
My friend, ha'Rav Yehudah Landy of Telz-Stone, suggested that Rashi perhaps tried to avoid that explanation since the Gemara tells us (Zevachim 19a) that according to Rebbi Yochanan, Yitur Begadim only applies if an extra garment is worn on a place which is normally covered by another garment . A string or ribbon tied around the hands is not considered Yitur Begadim at all, according to one version of Rebbi Yochanan, or is only considered Yitur Begadim if it is 3x3 finger's widths, according to the other version. Yisachar's cloth may not have qualified, then, as Yitur Begadim.
2) With respect to the question of which animal--sheep or goat--is superior, were the king and queen talking about asthetics of appearance or taste? (There is a diyuk to be made in Rashi, I think.) Either way, what is the limud behind this strange chakira?
15) [line 8] GADYA YA'I - (a) a goat tastes better than a lamb (Rashi Pesachim 57a); (b) a goat is more fit to be offered as a Korban than a lamb (Rabbeinu Gershom on our Sugya)
According to Rabbeinu Gershom, obviously the question is which is the more honorable/kingly of cuisines.
As for the Limud -- besides the story of Yisachar's just punishment -- perhaps it is the following.
What was the difference between Yisachar and Yishma'el ben Fi'abi? Not very much, at first glance. Yishma'el dressed in uniquely beautiful Begadim (Yoma 35b). Yisachar wore beautiful silk gloves when performing the Avodah. Both seemingly showed the glory of the Kehunah. However, Chazal saw that Yishma'el's intentions were l'Shem Shamayim while Yisachar's were l'Shem Kavod Atzmo. Of Yishma'el they said "when Yishma'el ben Fabi died, the splendour of the Kehunah died with him." Of Yisachar they said, "the Azarah shouted to have him thrown out!"
(BTW, the same is true of Bnei Eli and Yochanan ben Nadva'i. Bnei Eli stuffed themselves with the Kodshim she'Lo l'Shem Shamayim, (Shmuel ! 2:14,15), while Yochanan filled his stomach l'Shem Shamayim.)
The Torah equates the offering of lambs to that of goats, even though lambs are generally larger and fatter ("Patruza bar Danka" refers to goat-offerings, Bechoros 11a), to teach us "Echad ha'Marbeh v'Echad ha'Mam'it..." the main thing is to have the right intentions . Yisachar, who thought outward appearances were the main thing, thought that a lamb was chosen for the Tamid for that reason. He was wrong. He didn't even learn the Scriptures properly, the Gemara states. The main thing is what is in a person's heart.
3) Comment: I have read (I forget where, of course) that in Near Eastern countries in which cutting off hands is practiced, the custom was to use the right hand for food and the left for personal hygiene. Losing one hand, therefore, imposed on the individual a very harsh existence, the details of which are, I think, self-evident upon reflection. Imagine then the condition of this Yisachar, who was not only rendered obviously posul for avodah, but whose entire existence was reduced to a miserable choice between trying to find someone who would feed him in his state of constant malodorousness.
4) And yet, Rav Yosef, could still find the wherewithall to bless hashem for having "dealt with" this Yisachar. Consequently, this by way of suggestion, althought he gemarah in pesachim does not do so, the gemara here ends off with the famous Talmidei Chachamim Marbim Shalom ba'Olam....