"Can you give me some bread?" is considered wrong whilst "Could you give me some bread?" (which in the wrong tense) is a correct solution as well as 'Can you give me bread?' (which I have no problem with)
There is no reason "est-ce que tu peux" cannot translate to "can you."
Yes, just like in English, you can ask a question different ways: "Can you give me some bread?" or "You can give me some bread?" The difference would be the context and what you are trying to emphasize.
And how does it look like in short version when you drop "est-ce que"? Is it going to be: "peux-tu ..."?
Est-ce que is generally used as way to avoid inversions while asking questions, and is slightly more informal. Yes, it can be dropped here if needed.
Est-ce que tu peux me donner du pain ? Je ne suis pas riche. S'il te plaît ! Ricaner
I think it's because the action that is being performed on "me" is "donner" and not "peux."
While the reason below is true, it is proper grammar to put the pronoun (whether it is direct or indirect) in front of the infinitive. ex. I would like to see you. : J'aimerais TE voir. It becomes more complex, but if you are interested, visit: http://french.about.com/library/weekly/aa032301.htm
je peux, tu peux, il/elle/on peut.. In this case the 2nd person singular form follows the 1st person singular form, I hope that helps!
"Est-ce que" means "Is it that" so the literal translation of the sentence is "Is it that you can give me some bread". I'm pretty sure it's just an option to include that part so you don't have to.
It is proper grammar to put the pronoun (whether it is direct or indirect) in front of the infinitive. ex. I would like to see you. : J'aimerais TE voir. It becomes more complex, but if you are interested, visit: http://french.about.com/library/weekly/aa032301.htm
"Can you pass me some bread?" is also a valid translation, yet it's rejected.
Also of note is that 'Would you' is considered the polite for of 'can you' in English. So depending on the context 'Would you' would be a better translation