The Bavli (Taanit 29a) teaches that the Temple was set aflame on the ninth of Av in the afternoon and it burnt until sundown on the tenth of Av. The Yerushalmi (Taanit 4:6) records that because of this certain Amoraim would fast supererogatorily on the tenth of Av as well as the ninth, while others would fast only for the night of the tenth. (This is the source for the common custom of extending some of the customs of mourning from the Nine Days onto the tenth of Av (OC 558).) The Magen Avraham (ibid. sk 3) records an additional practice of fasting only until noon on the tenth of Av. This fasting is Halakhicly connected to Tisha bAv (and not just a conveniently-timed ordinary vow of asceticism) as seen from the ruling of Magen Avraham (OC 557:0) and Arukh HaShuchan (ibid. :3) that one fasting on the tenth of Av recites Nacheim in his prayers.
I have the following questions about these supererogatory fasts:
Do they require acceptance at Mincha beforehand like regular personal fasts (OC 562:5)? If so, must the acceptance be made at Mincha on Erev Tisha bAv or on Tisha bAv itself?
What time intervals may they occupy? The Talmud recorded ~12 and ~24 hour additions, and the Magen Avraham recorded an ~18 hour addition. Can one accept to fast until midnight on the 10 Av (a ~6 hour addition)? Can one accept to fast for only the first hour of 10 Av?
At which prayers would Aneinu be said?
Ashkenazim generally recite Aneinu only at the last prayer on a fast day (generally Mincha) (OC 565:3). When fasting multiple consecutive days, Aneinu is said at the last prayer of each day (Terumat HaDeshen 180, Rama OC 562:7) since we take each day as a unit. How does this rule apply in our case: is Aneinu recited only at Mincha on 9 Av? At Mincha on 9 Av and the last prayer during the fast of 10 Av (whichever prayer that may be)? At Mincha on 9 Av and all subsequent prayers of the fast? At no prayer other than the last prayer during the fast of 10 Av (whichever prayer that may be)?
Sefardim generally follow the Ashkenazi position only on personal fasts, but on public fasts they recite Aneinu whenever fasting (OC 565:3). How does that play out in this case? Do these fasts have the status of decreed fasts as an extension of Tisha bAv, or are they separate unique personal fasts?