Subject-Verb Agreement, Imperative Sentences – Why Isn’t Third Person Singular Used in ‘The Lord Bless You’?

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").

Typically, only the first verb in each clause can be finite:

I play.
I played.
She plays.
She played.

As you can see, this verb changes form. It shows tense (present or past) and agrees with the subject (adding -s if the subject is third person singular present). This must be our finite verb.

Our example is a declarative clause. To make it into a question, we should turn it into an interrogative clause. And for this sort of sentence, we do that by swapping the subject and auxiliary verb, like this:

She was safe.　　→　　Was she safe?

This is called Subject-Auxiliary Inversion (SAI). We use SAI to mark these sentences as questions. And in this case, SAI is possible because we have the auxiliary be.

But what if our example doesn't have an auxiliary verb? Let's look at our example from earlier:

I play.
I played.
She plays.
She played.

We can't swap the subject and auxiliary verb if we don't have one! Let's solve that problem by adding the meaningless ("dummy") auxiliary do:

I do play.
I did play.
She does play.
She did play.

Now we've got two verbs! And only the first can be finite. In this case, that means do! And as you can see, do now changes form, but play doesn't! That's because play is no longer finite.

Now we can make it into a question. So, let's do Subject-Auxiliary Inversion again, moving the auxiliary before the subject I:

Do I play?
Did I play?
Does she play?
Did she play?

These questions are formed correctly. But what about your example?

*Does she plays?

It doesn't work! Why not? Because we'd have two finite verbs.

In this answer, the * symbol marks a sentence as ungrammatical, and bold in examples marks a verb as finite.