Any and some are sometimes hard in their non-literal, contextual usage. As you know, they literally just mean "a small quantity," but the social connotations--how you can use them to influence the impression that you give people--is not obvious. Unfortunately, it's also very hard to make generalizations, rather than look at a specific context.
If there is a general rule, I would say it is this: these are words that refer to a small quantity, not a large one. So these words can be used when asking for things (to imply that you just want a little--making the request seem less burdensome), or when talking about oneself (to appear modest). If you aren't using them just to indicate a quantity, you are probably trying to emphasize the modesty (sense 3) of what you're talking about.
Do you have any money?
This could appear in a number of contexts. Without emphasis, it implies "with you right now." That might be because I want to borrow some (why else would I ask you about your money? rude!), or because we're headed to a place where you might want to buy something and I want to be sure you brought it with you; the exact details would depend on our relationship.
Where I have seen "any" emphasized is in people who are asking for money without us necessarily having a relationship that would allow them to ask for money. So someone soliciting for a charity, or begging on the subway, might say "Please, if you have any money..." The stress on the word "any" emphasizes that they want me to think of the request as being small, not a burden on me; and also says that they will take anything they're offered, no matter how small.
Compare this to "Do you have money"? That's a completely different question: absent context, I would assume that someone was asking about my general financial circumstances--whether or not I was rich. Though again, the precise meaning would vary based on our relationship.
I hope I've been of some help.
I don't see nearly so big a difference between this and "I hope I've been of help" as there is in the previous example. I'm not sure that "some" even colors the meaning at all if it is not emphasized. If it is emphasized--"I hope I've been of some help"--it suggests that I, the speaker, realize I haven't been able to solve your problem completely, but I hope that I have at least improved it a little bit. That is to say, I hope this wasn't worthless.
Can I have some water?
Here "some" is there because otherwise it sounds like I'm asking for all the water in the world or something. We usually use some sort of measure word or quantity word when asking for a liquid, unless we literally want all of it that's available. So here "some" without emphasis is just replacing "a bottle of," "a cup of," "a drink of" etc.--some other more specific quantity. You might emphasize "some" if you wanted to stress that you want a small amount of water, where otherwise none would be expected; but that's rare and you shouldn't worry about it.
The only case I can think of where someone would use "Can I have water?" would be to choose among different available choices. So a child might say "Can I have milk?" if asking his parents' permission to drink milk; or you might say to a restaurant server "Can I have water?" if the server had offered you a soda or something. "Can I have water?" is a reasonable polite response to the question "What would you like to drink?" but would sound weird in response to "Can I get you anything?" If I asked what you want to drink, you're choosing from the beverage options. If I asked "Can I get you anything," you'd be requesting something specific, which would have to be some quantity of water, not "water" in general.
I know this is long and confusing--I hope that it has been of some help, nonetheless!