There are two questions here:
a two-day programme
a three-man committee
the use of the indefinite article refers to the words 'programme' and 'committee' which are singular, and thus allow (or sometimes require) the use of the indefinite article. Articles (and other determiners) are positioned at the beginning of a noun phrase, even though they refer to the head noun and not the words that directly follow them.
This is simply a way to construct this type of words (which are, as StoneyB pointed out, used for measurements). When you combine a number and a noun in this particular way you don't have two words any more, you have a new, compound word, which follows new rules. Compound words can be written unseparated (a bookworm, lifetime), with a hyphen (last-minute, sun-dried, two-day, three-man, etc.) or they can be written separately (world famous). With the type of compound words you are asking about, a hyphen is used.
Note that you would say:
I attended a two-day conference.
Which means that you attended a conference which lasted two days.
If you said:
I attended two days of the conference.
means that you were there for two days, but the conference may have (and probably has) lasted longer.
Or consider this example:
A two-word phrase is one that consists of two words.
Fund is a countable noun meaning an amount of money kept for a specific purpose. There can be many different funds.
Funds, in addition to its meaning as the plural of fund, is used as an uncountable noun synonymous with "money." Thus, it would take the comparative adjective "much" just as "money" does:
How much funds did the organization raise?
However, the construction is awkward, and would seldom be used by a native speaker. Instead, we would say "How much money?"
Funds takes the plural when it serves as the subject:
The funds are not sufficient to the task.
When all is said and done, funds, when used to connote money, is a plural uncountable noun.
Best Answer
Mostly singular, but it could be plural in some cases (depending on the intention of the speaker) and there is enough ambiguity that treating it as either singular or plural is unlikely to cause problems. Both singular and plural may be acceptable.
So what is going on here. Normally adjectives can precede nouns with no "of", so this means that the superlative has actually been "promoted" from an adjective to a noun. "The most intriguing" means "The most intriguing one", or possibly "The most intriguing ones". Here you can see the ambiguity in number. Both "one" or "ones" is possible as the intended meaning.
Here is a much simpler sentence that shows the same ambiguity in number.
A superlative adjective "best" is promoted to a noun. But it can be treated as either a singular or a plural noun. In this case, the number ambiguity is resolved by the complement. If the complement is John, then "best" means "best one" and is singular, etc.
But, in your more complex sentence, there is no way to resolve the ambiguity in the number of "the most intriguing". It could be either singular or plural. In the first case, the verb implies that it is actually plural, in the second it is singular. Singular is probably more common, given the usual meaning of superlatives.
The plural "efforts" is a modifier, and it doesn't change the number of the noun. But it's proximity to the verb can cause interference. Especially when, as here, the number of the main noun is ambiguous.