words - Can I help you?

In English, the word "help" can be used for any of these cases to ask somebody for help or to give a help to someone:

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  In a store, when a clerk (store worker) says:

  
  

  
  

  Can I help you, sir?

  
  

  
  


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  With friends, when you see that your friend needs some help on math:

  
  

  
  

  Do you want a help with math?

  
  
  

  
  


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  When someone is carrying heavy things:

  
  

  
  

  Shall I help you?

  
  

  
  


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  When it's me who needs some help.

  
  
  

  
  

  I want a help to make cookies.

  
  

  
  


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  Help with household things (housework) such as cleaning, ironing:

  
  

  
  

  Son, please help me cleaning the dishes.

  
  

  
  



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  To thank:

  
  

  
  

  Thanks for your help.
  I'm happy that I could help you somehow.

  
  

  
  

In what situations are the Japanese expressions "手伝う", "手を貸す", and "助ける" (or in the forms "手伝いましょうか?", "手伝ってくれ?", "手を貸して。", "助けてくれ。", :教えてあげる。", "教えよっか?") used?

Answer

I fully agree, this is much harder to translate (well) than one would expect.

There are hundreds of nuances and scenarios covered by the English "can I help you", and you list a lot of them... So I'll focus on three very typical broad categories (I'm sure people will give you more):

Strangers: Typically, offering your help finding directions to a tourist who looks lost.

Rather than a direct "Can I help you?", any variations on "are you OK?" (implying that you are ready to help) is probably the best way to go. Depending on context, anything from a 大丈夫ですか？ to 道に迷っているんですか。

Close friends and family: Where an equivalent of the informal "Need a hand?" would be appropriate. Then 手伝う can come handy... From a purely cultural standpoint, I still would try to keep it sounding more like an offer than a question. E.g. 手伝ってあげよう ("let me help!") rather than (the otherwise perfectly correct): 手伝って欲しい？ ("do you want me to help?").

Less close friends, subordinates or same-level colleagues would be variations of the above (with appropriate use of polite verbal forms).

Your boss (or any person high-enough above you): is a different matter. There are many sonkeigo-infused expressions to offer help. They all have in common that you must make it sound like you are asking for a favour, not doing them a favour.

One of my personal favourite sonkeigo expression for that is:

お手伝いさせていただきます

(with countless variants:)

お手伝いさせていただけますか

お手伝いさせていただきましょうか

手伝わせていただけますか

etc.

Which literally means you are begging your boss to do you the favour of accepting your help.

Edit: as for ways to thank somebody for their help. It is once again down to context. Most basic (and literal) way would be: 手伝ってくれてありがとうございました but if it was a big favour/help, you can't go wrong by focusing on "the trouble you've caused" (and apologising for it), in which case you'd use: 迷惑をかけて申し訳無い (for bigger favours) or a simple すみません (which means both "sorry" and "thank you" in that context).