Is it possible to have an interrogative after an imperative connected with coordinating conjunction in a compound sentence?

There is, apparently, no rule carved in stone; rather, what dictates the use of coordination by means of "and" (bar some fundamental and rather stringent requirements as regard to parallelism) is a general precept that is referred to in CGEL § 13.22, p. 930.

And is a coordinator which has the most general meaning and use. The only restriction on the use of and as a coordinator is the pragmatic one that the clauses should have sufficient in common to justify their combination. Thus the following is odd simply because it would be difficult to find any connection between the content of the clauses:

• The youngsters went off to a dance and the equator is equidistant from the two poles.
  (see this comment for a similar example)

The § 13.22 goes on as follows.

In logical terms, and merely conveys (for declarative clauses) that if the whole sentence is true, then each of the conjoined clauses is true. But the pragmatic implications of the combination vary, according to our presupposition and knowledge of the world. The relation connoted by the link between the two or more conjoins can generally be be made explicit by the addition of an adverbial. By using the term 'connotations', we intend to indicate that the relations of meaning between conjoins are not hard and fast: they vary in strength, and more than one can coexist in the same occurrence of and.

The section goes on to tell us that there are eight types of connotations and that "in only three — (c), (f), (g) — can the sequence of clauses perhaps be reversed without changing the relationship between the clauses".

The eight types
(a) CONSEQUENCE or RESULT
(b) SEQUENT
(c) CONTRAST
(d) CONCESSIVE
(e) CONDITION
(f) SIMILARITY
(g) ADDITION
(h) COMMENT or EXPLANATION

Examples (wherever possible, insertion of the adverbial making the relationship explicit)

(a) He heard an explosion and he (THEREFORE) phoned the police.
(b) I washed the dishes and (THEN) I dried them.
(c) Robert is secretive and (IN CONTRAST) David is candid.
(d) She tried hard and (YET) she failed.
(e) Give me some money and (THEN) I'll help you escape.
(f) A trade agreement should be no problem and (SIMILARLY) a cultural exchange could be easily arranged.
(g) He has long hair and he (ALSO) often wears jeans.
(h) There is only one thing to do now — and that's to apologize.

The two sentences fit case "(g)". However, they fail to comply to a fundamental requirement to which is subjected the use of coordination: the structures must be parallel (Creating Parallelism Using Coordinating Conjunctions).

• Please tell us the best choice, and why/why is that?
• Tell me your name, and where are you from?

What sounds unusual to the ear is the request for information through an imperative coordinated to a question in direct form. There are two solutions (variants of punctuation are possible, and possibly some other small changes).

• 1     Please tell us the best choice, and (ALSO) why that is so.
         Tell me your name, and (ALSO) where you are from.

• 2    Please, tell us; what is the best choice, and (ALSO) why is that so?
        Tell me, what is your name, and (ALSO) where are you from?

Is it correct to say:

Please tell us the best choice, and why/why is that?

Tell me your name, and where are you from?

No. The second clause of both should be expressed as an indirect question:

Please tell us the best choice, and why/why that is.

Tell me your name, and where you are from.

You do not need the question mark and the awkward inversion and unsalvageable punctuation goes away.

16/01/21 Edit in response to a further point by the OP:

Thanks. But I specifically asked what is the rule here that won't allow it. Is it parallelism?

In a way, "Yes."

You should understand that there are two types of question, direct and indirect.

How about two question connected with "and": "What's your name, and where are you from?"

That is fine - 2 direct questions joined by a conjunction.

"Please tell us what your name is, and where you are from." = That is fine - 2 indirect questions joined by a conjunction.

You have tried to join a request and a direct question with a conjunction, and you can't do that...

If you look at your question, it should be obvious:

Please tell us the best choice, and why/why is that? = Please tell us the best choice, and please tell us why/why is that?

In Please tell us X, if "X" requests an answer/response (which your example does), then "X" is always an indirect question or request. "Why that is" is an indirect question, so that works... "Why is that?" is a direct question, and does not.

The answer is no, generally speaking. And the explanation is rather simple:

You have two independent clauses—one imperative, one interrogative—that, when coordinated, fight for sentence-ending punctuation.

To better see the problem, just reverse your example:

Original: *Tell me your name, and where are you from?
Reversed: *Where are you from? and tell me your name.

In the reversed version, the interrogative clause demands a question mark, which immediately ends the sentence.

In the original version, the imperative clause inherits an unwanted question mark (*tell me your name?) from the whole sentence.

So keep the clashing clauses separate. Or better yet, use an embedded question construction:

Tell me your name and where you are from.

There, your name and where you are from are coordinated objects for the imperative tell: Tell me: 1) your name and 2) where you are from.