The Shulchan Aruch (565:3) writes that some say a yachid only says aneinu by mincha on a fast day because we are afraid that he will fall ill and will not be able to complete the fast causing his tefillah to be invalidated retroactively (but the shliach tzibbur will say it even by shacharis in the repetition because he can assume at least someone will complete the fast). The Shulchan Aruch continues by saying that this is only when communities fast for rain, but the four fasts that were instituted by the navi and accepted by klal yisroel one should say aneinu in all the tefilos because even if one eats it is appropriate to say aneinu since it is a yom ta'anis for klal yisroel. The Rama disagrees and writes that the minhag is always to just say it by mincha, even on the four fasts. Based on the logic of the Shulchan Aruch, on the four fasts one should say aneinu even in the ma'ariv preceding the fast [sha'ar hatziyun 8 explains that although one may still eat, the status of ta'anis begins even in the evening, which is clear from the fact that if one fell asleep they would not be able to eat when they woke up].
In the gemara it is clear that even when one is fasting a ta'anis for rain, aneinu is said in all the shemoneh esrei of the day, starting with ma'ariv the night before. However, Rashi adds that we don't say aneinu by ma'ariv and even shacharis because we are concerned that the person won't be able to finish the fast and will retroactively be considered a שקרן בתפלתו. The Maharsha explains that one cannot ask that if this were truly a concern, why was it only instituted by the ge'onim and not in the time of the gemara, because in the time of the gemara people were stronger and assumed to be able to complete the fast. Whereas in the time of the geonim people became weaker and this became a concern. According to this approach, one can only say aneinu in their shemone esrei if that day has the status of a ta'anis, which is achieved by fasting for the full day. The machlokes between the Shulchan Aruch and Rama is whether the four fasts automatically assume the status of ta'anis based on chazal takana, regardless of the hanhaga of the individual.
Tosafos asks on Rashi that if by not completing the fast, one essentially ruins the tefillos that they davened with aneinu, why does the gemara allow for one to break their fast midway through, and make up for it a different time. By breaking the fast they are invalidating their tefillos that were said with aneinu? Tosafos asks this as a question, but the Ran says that based on this question we see that the premise is wrong. Even if one doesn't end up completing the fast, they are not considered to be a שקרן בתפלתו because at the time that they davened they were fasting. Both the Shulchan Aruch and Rama assume like Rashi, that aneinu is only appropriate if the day will have a status of ta'anis, and not if one just happens to be fasting at the moment they are saying it. Furthermore, the Ran's approach doesn't explain why one would say aneinu by ma'ariv since he will be eating both before and after, it only explains why one can say aneinu by shacharis when he is actually fasting, even though he doesn't intend to complete the fast.
In the Mishna Berura 566:14 he cites a machloes whether a tzibbur can read va'yichal on a ta'anis when there isn't a minyan fasting (if it is monday or thursday in the morning they certainly can since it is a day for krias ha'torah anyway). The Eliyahu Rabba and Pri Megadim write that without a minyan fasting one cannot read vayichal even on a ta'anis tzibbur, whereas the Sha'arei Teshuva writes that on a Ta'anis Tzibbur even if only seven people are fasting the tzibbur reads va'yichal. This is all by a ta'anis tzibbur for rain, but it seems that reading vayichal when no one in the tzibbur is fasting on one of the four ta'aniyos should be dependent on the machlokes Shulchan Aruch and Rama. According to the S.A. since the day has status of ta'anis, vayichal could be read even with no one fasting, whereas according to the Rama it should have the same din as ta'anis tzibbur for rain that we need either 10 fasting or at least 7.
Similarly, the Shulchan Aruch 566:6 writes that one shouldn't get an aliya on the ta'anis unless he is fasting and intends to complete the ta'anis. This should also be dependent on the S.A. and Rama, according to S.A. one should be able to get an aliya even if they are not fasting, but according to the Rama they should not get an aliya unless they are fasting.
R. Akiva Eiger (24) has an interesting teshuva where he is me'supak whether the reading for mincha on Yom Kippur is a din ta'anis or a din yom tov, and therefore questions whether a choleh who had to break his fast can get an aliya. R. Akiva Eiger definitely assumes that the reading of vayichal on one of the four fasts which is a din ta'anis may only be given to one who is fasting, which is consistent with the opinion of the Rama.
However, the Chasam Sofer (O.C. 157) has a teshuva where he discusses anticipating an aliya for mincha on a tisha b'av that he wasn't fasting and questions whether he could have taken the aliya. In his second argument which he seems to imply is valid by itself is that tisha b'av is a takana as a day of ta'anis on klal yisroel, and therefore even one who isn't fasting is still obligated in the krias ha'torah. - והכא נמי ט' באב יום מועד דפורענותא הוא ואפילו אינו מתענה בו מ"מ מחוייב בקריאת היום
The difficulty with the chasam sofer is that his logic seems to be very similar to that of the Shulchan Aruch in that the day has status as a fast day, even if the individual is not fasting. But it is on this very point that the Rama disagrees and holds that without fasting, one cannot say aneinu, and a tzibbur not fasting should not be able to read vayichal, and an individual should not be able to get an aliya. How can the chasam sofer pasken along the lines of the Shulchan Aruch, against the Rama? Perhaps even he only says this for tisha b'av because it is a day of פורענותא, but on the other ta'aniyos he would agree that one who is not fasting cannot get an aliya. But, Rav Moshe (o.c. ?, 113) had a teshuva discussing this issue and definitely assumes that the logic of the chasam sofer would apply to all 4 ta'aniyos.