A question within a sentence should be preceded by a comma, and end with a question mark.
I am wondering, how long has that fish statue been there?
There are three important issues that this committee must address. What was the chairman doing in that YouTube video? is not among them.
If a question is particularly long or complex, then for clarity it could begin with a capital letter.
In the daydreams of my youth I would look at the clouds and wonder, Will I find happiness and fulfillment soon after getting a business administration degree, or must I amortize my well-being return on investment over the next seven to ten calendar years?
The sentence in the original question ends with a subordinate clause that does not, by itself, ask a question (“within which we can leverage our existing knowledge”). This is modifying a noun phrase that is part of the question (“set of simple computational rules”). Therefore, this subordinate clause is part of the original question, and should end with a question mark.
Putting all of this together, I would punctuate the sentence like this:
The question nagging me is, To what extent can we encapsulate all what we observe in nature into a set of simple computational rules, within which we can leverage our existing knowledge?
(Some of this advice is based on The Chicago Manual of Style 15th Edition §6.55 and §6.71.)
The "What's new" example you provided is not necessarily a question, it can be a statement as it was something like "Here you can find what is new", so it doesn't obviously need a question mark. Of course you can also name the section (of a site, for example) as "What's new?" and in that case it's legit to use the question mark.
Regarding Rhetorical questions, they would need the question mark because they are questions. Still, the page that I linked explains that they can be followed also by a full stop or an exclamation mark, depending on the context where they are being used.
See for example:
- Wasn't that game incredible!
- Why are you so stupid?
Best Answer
The question mark (?) is almost never used in English as a strictly syntactic marker.
You would do better to think of it as imposing interrogative intonation—the characteristic rising tone at the end of the utterance which 'invites' your hearer to respond. It can thus be employed with utterances which have the form of a declaration:
By the same token, the question mark can be omitted with utterances which have the form of a question: