A simple rule of thumb is, when you're talking about a noun that has a restrictive clause on it, use 'the'; if you're talking about an unrestricted noun, don't use 'the'.

An "unrestricted noun" talks about an entire group; a noun with a restrictive clause talks only about some subset of the group.

In your first example, "Americans" is an unrestricted noun: it refers to every American citizen. So, you can leave off "the":

22% of Americans want Bruce Springsteen to write a new national anthem.

In your second example, the noun is "times", but it has the restrictive clause "[that] I lie down"; we are not talking about all times, only a specific subset of them. So we want to use 'the' in this case:

25% of the times I lie down, I fall asleep.

Of course, having laid out this rule of thumb, I feel compelled to point out some exceptions. First, when you use a noun that is understood to be part of some larger group, even though it's not explicitly stated, you still use 'the':

Everybody here is hungry, and most of us have agreed to order pizza, but 40% of the Americans are holding out for cheeseburgers.

Secondly, time almost always takes 'the':

How often do I fall asleep when I lie down? Oh, about 25% of the time.

This is probably because it is implicitly restricted by the conditions in the question, although those conditions are not repeated in the answer.

(The use of all, while not strictly incorrect, is redundant and unnecessary.)

I basically agree with Hellion, but let me attempt to clarify part of his answer.

"Americans" refers to Americans in general, that is, all Americans. "The Americans" can refer to all Americans as a group. Like you could say, "The Americans are a hard-working people". But more often if someone says "the Americans", it's because he's distinguishing a portion of some larger group that are Americans from other members of the group. Like one might say, "Britain, France, and the U.S. sent representatives to a meeting to discuss this issue. The British and French were in favor of the proposal, but the Americans were against it." Here, "the Americans" means that, of the people who were at the meeting, those who were Americans took this position. Similarly in this sentence for "the British" and "the French".

This is not limited to nationalities. If you were talking about a group containing both men and women, you might say, "The women wanted to ... but the men wanted ..." Etc.

On a different part of the question: "All" is generally not needed in a context like this. It can be used to clarify, if you are switching between talking about some subset and then talking about the whole group again. For example, suppose you took an opinion poll and broke out the results by party affiliation. You might say, "70% of Democrats believe that ... but only 30% of Republicans ..." Now suppose you want to talk about the Democrats, and then about people of all parties combined. You could say, "70% of Democrats believe that ... but 45% of Americans ..." But now it sounds like "Democrats" is the opposite of "Americans". (Maybe it's the opposite of "true Americans", but that's a different subject. :-) ) A reader might reasonably be confused, thinking perhaps you're talking about some group in another country that is called "Democrats", etc. They'd probably figure out what you meant, but if the terms were unfamiliar to the reader -- like if you were talking about factions in ancient Rome or something -- he might not be sure of your meaning. So it is more clear to write, "70% of Democrats believe that ... but 45% of all Americans ..."

Both are correct in your example; the "the" is redundant and may be omitted.

Typically, "the" is used to clarify the meaning the word that follows it.

Replace "Americans" with "French" and the "the" becomes necessary, because "French" could be referring to the people or the language.

22% of the French... vs 22% of French.

22% of French doesn't really make sense, and would need further clarification such as 22% of French speakers.