I don't have a clear answer for the ultimate derivations. Here is what I've found so far.
General counting forms
First, a note regarding the counter つ forms: all of the geminate (double-lettered, with the small っ) ones are later developments, probably as a simple phonotactic emphasizing of the つ. むっつ was originally むつ, which is 六【む】 + counter つ.
The potential oddball here is 五【いつ】つ. This shows up in some sources simply as 五【い】つ, and in old compounds the い reading does appear, as in 五十【いそ】 or 五百【いほ】. But then we also encounter the reading いつ in other contexts, such as 五衣【いつぎぬ】 or 五重【いつえ】. From what I've read and understood, I'm inclined to view いつ as the underlying form, with い a derivation from that.
Now to look at the date forms. My source for etymologies is Shogakukan's 国語大辞典【こくごだいじてん】
- 一日【ついたち】: Odd, but explainable: suppletion, from 月【つき】 + 立【た】(ち), as in the start of the month.
- 二日【ふつか】: Odd. We'd expect ふた + か.
- 三日【みっか】: Mostly regular, gemination arising from older みか reading.
- 四日【よっか】: Mostly regular, gemination arising from older よか reading.
- 五日【いつか】: Regular.
- 六日【むいか】: Odd. From older むゆか reading, which is also odd. We'd expect 六【む】 + 日【か】, but there's this unexplained ゆ.
- 七日【なのか】: Odd. From older なぬか reading, which is also odd. We'd expect なな + か.
- 八日【ようか】: Odd. From older やうか reading, which is also odd. We'd expect や + か.
(The shift from やう to よう is a regular historical process.)
- 九日【ここのか】: Regular.
- 十日【とおか】: Regular. From older とをか reading, where とを is explained as an apophonic form of 撓【たわ】 "saddle or pass in a mountain range; bend, dip", as in 撓【たわ】む, from the way that one counts 10 in traditional Japanese one-handed finger counting by bending all one's fingers.
- 二十日【はつか】: Odd. We'd expect はた + か.
So ultimately we have five unexplainedly odd forms, all of which involve an extra
- Two where た changes to つ: ふた、はた > ふつ、はつ
- One where ゆ appears from nowhere: む > むゆ
- One where な changes to ぬ: なな > なぬ
- One where う appears from nowhere: や > やう
It's possible that all four patterns are from an insertion of う, causing related sound changes. However, that doesn't explain the
/j/ glide in むゆ. That might be a simple excrescence, such as the
/w/ glide added by some speakers in the term 場合, read as either ばあい or ばわい.
However, that is only speculation, and I have not found any research suggesting this is a linguistically valid hypothesis. Even if we accept this suggested う insertion, I have no idea what this う might have meant.
From the best I can find, the odd
/u/ forms above are only found in these date terms. I cannot dig up any other instance of 二 read as ふつ, 七 as なぬ, etc.