Some Questions

When you go to sightseeing sites, you must one may come across to see other tourists taking pictures.

"Would you like me to take a picture with your camera?”

This seems the most polite approach.

"Do you want me to take a picture with your camera?”

a more informal gesture, but right nonetheless.

"Do you need a help?" "Do you need some help?"

"Do you need my hand?"

Usually it is lending someone 'A' hand, so the question should be , do u need a hand?

Or is there any other phrase do you would rather say?

By the way, there are some English phrases that you can express the places where a lot of many tourists visit.

lot of and many have the same significance.

Which phrase do you use or hear the most?

Tourist spots , and i avoid them like plague :D

When you go to sightseeing sites, you must come across to see other tourists taking pictures.

Normally, we call "sightseeing sites" tourist attractions (ie, things which attract tourists).

And when you say that you come across someone (or you encounter them), you are implying that you have seen them, and so "to see" is not necessary.

They seem to want someone to take their pictures with their camera.

Although they might want more than more than one picture to be taken, the expression is still "Can you take our picture?" or you could write it as "Can you take some pictures of us?"

Again, when you say that you will take their picture, it is assumed that you are using their camera, and you don't need to include "...with their camera."

"Would you like me to take a picture with your camera?”

Would you like me to take your picture?

"Do you want me to take a picture with your camera?”

Both of these ways are fine, except the first is more polite.

Also, this expression "do you want..." as opposed to "would you like..." can be used if you are annoyed with someone or angry, but it depends on how you say it. "Do you want me to take a picture, or not!?"

"Shall I take a picture?" It sounds weird to me.

It sounds weird, because it is! It sounds like a question that would be asked a hundred years ago!

I think normally, people use "should" as opposed to "shall" in everyday language.

"Do you need a help?" "Do you need some help?"

And again, if you want to be more polite because these people are older, you could say "Would you like some help?"

"Do you need my a hand?"

When you say "Do you need my hand," it literally means "Do you need my hand." Although I'm sure people will understand you.

The only other time I can think of when you ask for a hand, is when you are asking someone to marry you. "Will you take my hand in marriage?"

Which sentence do you think it sounds/looks more natural?

Or is there any other phrase do you would rather say?

By the way, there are some English phrases that you can use to express the places where a lot of many tourists visit.

Which phrase do you use or hear the most?

Tourist attractions. Sightseeing is what I would use more as a verb, or you will see "sightseeing tours" which take you around an area showing you all the tourist attractions, and maybe explain some history.

"Would you like me to take a picture of you with your camera?”

"Do you want me to take a picture of you with your camera?”

"Do you need a help?" "Do you need some help?"

"Help" is not a count noun. It will never be preceded by "a".

"Do you need a hand?"

This is an idiom. "A hand" is equivalent to help. Even though it is "my hand" I will literally use to provide the help, one does not say "my hand" in this expression.

Which sentence do you think is most natural? [Alternative: Which sentences do you think are more natural?]

If you are asking for a single sentence which is more natural than all of the rest, that is the "most natural" sentence. If you are asking for several sentences which are more natural than the others, then you need to use the plural "sentences".
Both of the first two are equally natural. "Would you like..." is more polite. They are the clearest ways of indicating what help you are offering. If it's clear what you are offering to do, "Do you need some help?" and "Do you need a hand?" are fine, and can be shortened to "Need some help?" and "Need a hand?"

Or is there any other phrase you would rather say?

You have already made this a question by asking "is there". You would only use the "Do you... say" construction if "say" were the main verb of the sentence, as in: "Do you say other phrases as well?"

By the way, there are several/many English phrases that you can express the places which a lot of many tourists visit.

Use "a lot of" or "many". The two are not used together.

Which phrase do you use or hear the most?

It depends on the context.

"Shall I take a picture?" It sounds weird to me.

It isn't weird at all! It's pretty common to hear "shall" spoken in the UK. It makes a request or a question sound more polite.