Both statements are correct, though they have different emphasis. When you say:
Please! Call me.
You're simply asking someone to call you. If you add "do", making it:
Please! Do call me.
Then, there's an implication that the other person may not call -- they may be forgetful or unwilling -- and you're emphasizing that you want them to make sure they call you.
I speak American English, so I'm going to substitute "will" for "shall", since it's more common in American English. All the following statements are correct and have essentially the same meaning:
Yes, I will.
Yes, I'll call.
The three answers may vary slightly in terms of formality, but it's not a significant difference (shorter answers and answers with contractions are ever so slightly less formal).
Only the first sentence is correct:
Does he do all those things?
When constructing a sentence starting with "Does he/she...", you use the base form of the infinitive (e.g. "to do" -> "do").