The answer to all of these questions seems to be that on Yom Kippur the act of washing per se is not what is assur; the act of anointing per se is not what is assur – what is assur is being mevateil inuy, of getting pleasure from these acts. It’s the end result, the enjoyment, which is assur -- not to act per so. So, for example, if someone has to put cream on because of a skin problem, that’s not a bitul of inuy, as the purpose is medicinal.
The Ran goes a step further asks how it is permissible to bathe a child on Yom Kippur when we know there is an issur of giving ma’achalos assuros to a child. If inuyim are derabbanan, there is no problem because the Ran (and Rashba in Yevamos) assume that the issur of “lo ta’achilum,” of giving a child something issur, only applies to issurei d’oraysa, i.e. the Ran holds you cannot feed a child neveilah, but you can feed a child tevel derabbanan (other Rishonim disagree). But if inuyim are d’oraysa, how can you wash a child?
The Ran limits his question to the inuyim of washing and anointing, but does not question the fact that a child must be fed –- apparently Ran holds that fasting would be dangerous, and even an adult is not obligated to jeapordize one's life to fast. However, Rashi seems to disagree. When the Mishna (Yoma 82) writes with respect to little children that that “ain m’anin osam l’sha’os,” Rashi explains that there is no *obligation* to withhold food even for a few hours. The implication is that the choice is left to the adult whether or not to impose restrictions on the child, something that we certainly would not say if there was any danger involved. Rashi, contra to the Ran, similarly makes very clear that when the gemara discusses which inuyim a child is allowed to violate and why (Yoma 78), eating is very much part of the discussion -- it does not fall under the blanket heter of sakanah.
Even those Rishonim who hold washing and annointing are only derabbanan and therefore children are exempt hold that eating and drinking are issurei d'oraysa. Therefore, if, as Rashi holds, there is no danger in a child fasting, why is one allowed to give them food? Why is there no issur of "lo ta'achilum" like there is by neveilah? How do you answer the Ran's question?
The Divrei Yechezkel (15:19) comes up with a unique chiddush here that needs one bit of background to get. The gemara in Kiddushin (34) gives a few examples of mitzvos aseh that women are obligated in because they are not zman gerama, e.g. ma’akah, hashavas aviedah. Tosfos asks: of all the examples to choose, it seems that in the one’s the gemara gives it makes no difference whether women women are obligated in the mitzvas aseh or not because in all of these cases there is a lo ta’aseh involved anyway. If you don’t build a ma’akeh, there is a lav of “lo tasim damim b’veisecha;” if you don’t return a lost object, you violate a lav of “lo tochal l’hisalem.” Since women are obligated in lavim, l’mai nafka minah in these cases that they are also obligated in the aseh?! Tosfos comes up with a technical answer, creating various scenarios where the aseh applies and there would be no lav. Ramban, however, writes a yesod: sometimes a lav does not stand on its own, but comes simply to bolster a mitzvas aseh. The lav of “lo tasim damim” exists only because of the aseh of ma’akeh that drives it; the lav of not returning an object is there only because there is the driver of the mitzvas aseh that compels action. The lav and aseh are not two independent forces – there is only one driving force and that is the aseh; the lav just tags along.
So too, suggests the Divrei Yechezkel, when it comes to the prohibitions of Yom Kippur. They are all there just there to bolster the mitzvas aseh of “t’anu es nafshoseichem,” but if you don’t have the aseh, there can be no lavim. Why can you feed a child on Yom Kippur but you can’t feed him/her neveilah? Because the issur of “lo ta’achilum” prohibits giving the child ma’achalos assuros like neveilah, but “lo ta’achilum” doesn’t mean you have to chase after a child to take a lulav or put on tzitzis or do other mitzvos aseh. Since inuyim all stem from a chiyuv to fulfill a mitzvas aseh, there is no chiyuv on an adult to force the child to do anything.
To end off with a question: what would be the din if someone is oseik b’mitzvah on Yom Kippur – does the rule of oseik b’miztvah patur min hamitzvah remove the obligation to fulfill the aseh of “ta’anu es nafshoseichem” and cancel m’meila all the inuyim that go along with it or not?