Under what conditions is one permitted to skip a minor fast?
Fast days are not optional. The "minor" means that they start at daybreak rather than the night before (as do tish'a b'av and yom kippur). Note the rules quoted below for when one may skip the fast (because of illness).
Yom Kippur is required by the Torah and Ta'anis Esther is connected to Purim not the destruction of the temple. Once the temple will have been rebuilt, the requirements will change, but that does not effect what we do nowadays.
There are circumstances in which one may "vow" a fast on the following day as part of the mincha prayer but that is not the case of the fasts that you were asking about.
ArtScroll footnotes 10 and 11 to Rosh Hashannah 18b1
Ritva concludes that according to all views, these fasts have been accepted by the nation and neither a community nor an individual has the right to disregard them. See also Shulchan Aruch, Orach Chaim 550:1
The following rules apply to all fast days aside from Yom Kippur and Tisha B’Av, which have their own rules (see our Yom Kippur and Tisha B’Av guides).
If you are pregnant or nursing and are in pain or feel weak, do not fast on this day. If you’re ill, consult with a rabbi. But even if you are exempt from fasting, skip the delicacies and sweets for a day.
There are five minor fasts on the Jewish calendar. With one exception, these fasts were instituted by the Sages to commemorate some national tragedy. The minor fasts (that is, all fasts except Yom Kippur and Tisha b'Av) last from dawn (first light) to nightfall (full dark), and one is permitted to eat breakfast if one arises before dawn for the purpose of doing so (but you must finish eating before first light). There is a great deal of leniency in the minor fasts for people who have medical conditions or other difficulties fasting. (emphasis mine)