How to apologize sincerely and how to politely ask others to give a hand?

There is no way to use words to create sincerity. Someone can say all the correct words in an apology, and still not be sincere. You sincerity, if the other person or people are able to detect it, will come from your demeanor, not the words you use.

When asking anyone for a favor, such as assistance, "please" and "if you please" should be sufficient.

I am always told to write notes to apologize sincerely. I am sorry is far to enough. If you please to tell me other phrases to say sorry.

I think you mean 'I am sorry is not enough', or 'I am sorry is far too little'.

It sounds like you are asking about formal written apologies, which are not always sincere. To English speakers, in general simple utterances are felt to be more sincere than complex ones. Merely saying you are sincere is not evidence that you are sincere. You can increase the amount of apology expressed by prefacing 'sorry' with one or more words like sincerely, very, deeply, profoundly, utterly etc.

Yes, I mean I am sorry is not enough. But I think 'I do apology 'is really weird. Can you tell me other phrases? Also, other phrases to say thanks.