Moed Kattan 19a - Avel: Difference Between Yom Tov and Shabbos

The Mishna says that Shabbos counts towards shiva but does not stop it, whereas Regel stops it but does not count toward it. The simple understanding as pointed out by Nimukei Yosef is that Regel has simcha associated with it, which is more powerful than oneg of shabbos.

Perhaps the mishna can be explained based on another line of thinking. The mishna indicates that one day yom tovim such as shavuos, r"h and y"k may have a "shabbos" status rather than regel. The nimukei yosef explains the rationale is that regel should really be no better than shabbos, meaning it should count toward shiva but not stop it. But, since pesach and succos are 7 days, even if they count toward aveilus they will end up stopping it. Based on this logic, even the chachamim may hold that we don't want to distinguish between one regel and another so we allow even shavuos to stop the aveilus. According to this line of thinking shabbos and Regel are exactly the same (except for the private things such as relations and studying torah, that is assur on shabbos but is mutar on regel). If aveilus began before then shabbos and regel "count" but by regel (pesach and succos) that translates into stopping the shiva (and lo p'lug for shavuos). But, if the burial takes place only during the regel then the regel doesn't even count toward shiva and the same would be true by shabbos, just that there is never a burial on shabbos.

However, regarding regel stopping the sheloshim, the gemara implies that the logic is totally different. Really regel should count toward 30, but not stop it. But, since we allow grooming for kavod haregel on erev yom tov, "ho'il v'hutra hutra" - once we allow violation of sheloshim, we don't want to go back and assur it. That is why rav and rav huna insist that one must take advantage of the heter to shave in order to abolish the sheloshim, whereas r' sheishes/abah shaul would say that even if you didn't take advantage, since it was mutar we don't go back and assur it after the regel.

Moed Kattan 15a - Mourner Wearing Tefillin

The gemara says that an avel does not wear tefillin. We pasken like R' Yehoshua on 21a that he only abstains from Tefillin on the first day. This is paskened in Shulchan Aruch Y.D. 388 and O.C. 38:5. It is not clear from the gemara and Shulchan Aruch if this exemption only applies when the day of burial is the day of death (which is d'oraysa aveilus acc. to behag and rif), or even if the burial is a day or 2 later we still exempt the avel from tefillin on the day of the burial. The Pischei Teshuva in Y.D. brings 2 opinions about this, and the M.B. is machria that we always exempt the avel on the day of burial from wearing tefillin even if it is not the day of death. This is clear from the Rosh. The Behag and the Rif are of the opinion that aveilus is d'oraysa on the first day, when the burial and death occur on the same day. Based on this it could be that the exemption of tefillin would only apply on the first day if it is "d'oraysa" meaning the yom hamisa and yom hakevura. However, the Rosh writes (end of siman 3) that the exemption from tefillin is even for aveilus d'rabonon. The Rabbonon felt that it was inappropriate to wear the crown of Tefillin while in a state of depression, therefore they uprooted the mitzvas aseh of tefillin on the day of the kevurah. It is for this reason that until the burial the avel does not put on tefillin (or do any positive mitzvah) bec. he is an onen, but even after the burial he does not put on tefillin later that day.