Yes, that sounds natural, but it should be: "Is there any way I can help you?" You can also say, "Can I help you?" Or "Can I help you with anything?"

Is there anything I can help you with? I think it'll be better if you ask in this way.

Yes, that is fine. Other options are as follows: "Is there some way I can help?" "What can I do to help?" "How can/may I help you? (Most often seen in the service industry). Incidentally, if you are speaking, what you wrote in your question is fine. No one will be able to tell the difference. However, if you are writing, you should write "Is there any way that I can help you?" "Anyway," the way you wrote it means "even though"/"still"/ "regardless"/"however." It is also a way of changing the subject, or returning to a previous topic of conversation, kind of like "bien" in French/Spanish, or "pues" in Spanish. Sorry, those are the only two of your languages that I know.

Here you mean 'any' to describe 'way' (ie. 'some kind of way'), so it'd be two words. Apart from that, this looks pretty good.

Yes, it does, but the correct spelling is "is there any way I can help you?" Of course, however, like all sentences, whether it is completely natural also depends on the context. If you're a salesperson, for example, and you see a customer appearing to look for something, it might be more natural just to say "may I help you?" On the other hand, in the context of talking to a friend who experiencing some personal problems, and after discussing several dead-end proposals to help that friend, you may finally say, "is there any way I can help you?"