Learn English – “I theself Naresh” as an introduction

The difference between "I and my friends" and "my friends and I" is purely a matter of courtesy - they are both grammatically correct. I would tend to stick to the latter though, as it a) is more commonplace, b) is considered more polite, c) seems to flow better.

Indeed, your example of 'incorrect' usage is incorrect solely in that the first sentence uses the accusative (objective) pronoun me, when you actually need the nominative (subjective) pronoun 'I'. The second sentence of that example is correct, since the pronoun needs to be in the accusative, as the object. You seem to understand this though; this is just to clarify.

The use of "that" in a question isn't right, and "have you" should be reversed.

You'd use "that" if it was a statement, not a question: "it is true that...", for example:

• I would like to tell you that you should have received my gift
• I would like to show you that you must have received my gift
• I would like to inform you that you will soon have received my gift

Your example is a question, so the word needs to match the intention of the question - indicating that this question seeks a yes or no answer:

• I would like to ask you if you have received my gift?
• I would like to ask you whether you have received my gift?

Replacing this changes the meaning of the question:

• I would like to ask you how you have received my gift?
• I would like to ask you when you have received my gift?
• I would like to ask you why you have received my gift?
• I would like to ask you who has received my gift?

Or maybe you meant more like this:

• I would like to ask you a question: have you received my gift?
• I would like to ask: have you received my gift?