Note: I'm posting this as answer rather than adding it to the question for two reasons: (1) I came up with this theory after I posted the question and after I read the other answers (2) adding this to the question would significantly change the nature of the question and it'd be unfair to those who already answered
After reading answers from Derek and Ross, I came up with a theory on how 度 and 回 differ from each other. Even though this theory came from my observation and lacked reference, the more I think about it the more probable it becomes so it might be the case that it is actually true.
Looking at the kanji for 回, it's the same kanji that is used for 回る, 回数, 回転 etc, which all imply a circular flow, a rotation, a repetition. This nuance of repetition is what 回 has and 度 lacks. The notion of repeating event brings about another nuance: intention (if you are causing the event to happen) and expectation (if you are just observing it). When an event is known to repeat, there is an expectation that it will happen again, while an event that simply occurs may not be expected to repeat.
If this is actually true, then it would explain Derek's answer why 度 becomes uncommon after the 4th time (or whatever the vague limit is). Events may occur the first time, second time and third time, but after the fourth time it is hard not to expect it will happen the fifth time. The fifth time is no longer "yet another occurrence"; there is an expectation for it to happen; the fifth time and so on are now repetitions.
On the other hand, 度 which is also used as a counter for degrees, while may not signify the notion of repetition, it has the notion of progression, i.e. that we do not only look at the occurrences as events that happen again and again but also compare the quality/quantity of the occurrences among each other. While 回 focuses on repetitions, 度 focuses on progressions. This can be seen from the subtle difference between 「今回」 and 「今度」.
This difference between 度 and 回 also extends to the choice between using 「もう一回」 and 「もう一度」 in a request. Based on my observation, 「もう一回」 in a request is loaded with intention and expectation that the requested action will happen, while 「もう一度」 lacks that expectation so it makes the request rather neutral and reserved. This probably explain why 「もう一回」 sounds a bit childish and is commonly used by children due to its bluntness in showing how they expect and insist the request to be fulfilled. On the other hand, the notion of progression in 度 loads 「もう一度」 with a request to improve the action. That's probably why when we want to ask somebody to repeat what he says because we couldn't hear the first time, it's more natural to use 「もう一度言ってください」 because we want the speaker to say it better/louder. 「もう一回言ってください」 may sound like we want the speaker to simply repeat exactly what he said the first time.
At this point, let's further extend this theory to distinguish the example sentences in my question. I think it's no farfetched to say that the difference between using 度 and 回 in saying "I haven't done it even once" lies in your expectation and intention; whether you want to do it or not):
I haven't done such thing even once (and I want to do it).
I haven't done such thing even once (and I do not even want to do it).
Any comment on this theory of mine?