... and your tone of voice is important. If you say it in a friendly manner, with a smile and eye contact, it wouldn't be seen as rude.

In America, "no thanks" isn't rude as long as you say it in a pleasant manner. You could also say, "No, I'm good, thanks", or "No, i'm ok." Of course you can always say "No thank you" if you are worried about it. If you did need help, you can respond, "Yes, where do you keep the soap?" Or, "Yes, can you tell me/show me where the milk is?" Hope this answers your question.

"No, thanks" is correct. If you want help, then, "Yes, please." Sometimes it does not feel friendly, because sometimes the clerk is not being sincere. If you are in a store where there is a lot of shoplifting, they may just be saying "May I help you?" to explain why they are talking to you. What they are really saying is "We have noticed you and we are watching you." "No thanks" is still the correct thing to say, but it explains why sometimes the interaction feels awkward or unfriendly. Really, in this situation, the store clerk is being rude.

No, it's not rude at all. However, it's normal to add a reason or explanation: "No thanks, I'm just looking."

A really natural, polite option would be "That's OK, I've got it, but thanks so much for asking."