Learn English – How to describe a future event

Question 1

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.

Question 2

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.
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).

Question 3

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").

Please let me know when you have shipped them is unambiguous but requires the shipper to understand " when you have shipped them".

"Please notify me when they have been shipped" is unambiguous but requires the shipper to understand "when they have been shipped".

Please send a postal tracking number if available, or a shipping notice with ETA uses nouns common to international online retail, so it is more likely to be understood than any sentence whose meaning depends on the verb tense, or on the shipper's understanding a phrase such as "at the time".