I know that #3 and #4 are wrong. But I need help in explaining why they are wrong. Is there a specific grammar rule that can explain why they are wrong?
For the sake of clarity, let's rephrase your sentence into the declarative (statement) form. The interrogative (question) counterpart will be explained afterward.
Smoking is {allowed/allow}.
The important thing is that you should know the function of the verb BE (i.e., verbs like be, is, was.. &c.) The central function of the verb BE as an auxiliary verb is to mark a passive voice and a progressive aspect.
So, if you want to 'package' the information in the passive voice, you will need the verb BE and its complement, which is necessarily a past participle; for this reason, the bare infinitive (allow) is not possible.
Smoking is allowed (NOT
smoking is allow)
As explained earlier, BE can be a marker of progressive aspect. In the progressive aspect, of course, the progressive (-ing) form is required.
He is working (NOT
he is work)
As you can see, BE is very rarely used with a bare infinitive. However, there's nothing impossible in English. "All he did was help me" is perfectly idiomatic. (The fuller explanation on this is here.)
Now why is the last sentence ungrammatical?
When converting a declarative sentence into the interrogative one, you must invert the subject and the auxiliary verb.
is she beautiful? (cf. the corresponding declarative she is beautiful.)
However, when there's no auxiliary verb in the declarative sentence, the auxiliary DO (or sometimes called a do-support) is needed and the main verb must be in the bare infinitive form.
Does school allow smoking? (cf. the corresponding declarative school allows smoking.)
I think this question, as it is actually phrased, deserves more attention.
Bare infinitives are licensed in a few cases. Offhand, I know that we use them along with modal auxiliaries, certain verbs of causation and perception, and sometimes with the verb "to help" (which happens to license both the bare and the full infinitive forms). We even use them on occasion to mark the subjunctive mode. There may be other cases that, as a native speaker, I'd find easy to recognize but hard to recall.
The verb "to be" doesn't fall under any of those cases. If we want an infinitive here, we'll need to use the full form.
Sure. Why not? Sometimes, it even works.
Is smoking to allow?
The sentence above is grammatically correct. Despite that, I still can't make sense of it. As it stands, it's a question that should accept both "yes, smoking does allow" and "no, smoking doesn't allow" as legitimate, complete answers. We have no object for the verb "to allow", and we have no context that lets "smoking" represent the kind of thing that allows something else.
Is the school to allow smoking?
The sentence above is also grammatically correct. Even more importantly, I can make sense of it. A school is the kind of thing that can allow something else, and the gerund "smoking" makes a suitable direct object for the verb "to allow".
We often use infinitives and infinitive phrases to represent things like intention, expectation and purpose. We can understand the question above to mean "does someone intend that the school allow smoking?" or "do we expect that the school will allow smoking?" or maybe even "is permission to smoke part of the purpose of this school?"
Those questions are, of course, quite different from "does the school allow smoking?"
Does the school allow smoking?
In this example, we do have a modal auxiliary: "does". The mode that this verb denotes is indicative, which is the reason that we can use it without changing the meaning of the clause. We commonly use it either to add emphasis or to form question exactly like the one above.
Thanks to "does", we have a reason to use the bare infinitive "allow" -- a reason that we don't have in the questions that start with "is". On the other hand, thanks to "allow", we have a reason to use "does". When forming questions we want the first word of the verb construction -- an auxiliary -- to come before the subject. When the verb construction lacks an auxiliary, forms of "do" come to our rescue.
Nah.
The verb "to be" is a special case. It works well with other auxiliaries, but it doesn't work with the auxiliary "do". We can take a statement like "it is true" and change it to "it should be true", "it could be true" and so on. We can ask question like "should it be true?" and "could it be true?" However, we can't form "it does be true" and "does it be true?". Instead, we let "is it true?" stand as the natural form of the question, without a hint of do-support in sight.
On what basis would you expect "Is smoking allow?" to be possible? Grammar doesn't just consist of putting words together at random - you have to follow the rules and use the correct forms.
The rules for forming questions can be found in various places ( When to use Does and Is while starting the Interrogative sentence? ; https://learnenglish.britishcouncil.org/en/english-grammar/clause-phrase-and-sentence/verb-patterns/verbs-questions-and-negatives ; https://www.englishgrammarsecrets.com/questions1/menu.php ).
Where you have an auxiliary verb, you form the question by inverting the order of the subject and auxiliary from Subject-Verb to Verb-Subject. Also, where you have the verb "be" in any of its forms, you form the question by inverting the order of the subject and "be", even if "be" is the main verb.
Consider your original sentence pre-inversion, i.e. the declaratory equivalent sentence. This is your example #1: "Smoking is allowed."
Inversion makes this into #2: "Is smoking allowed?"
It would be impossible to say "Smoking is allow", so the inverted form (#3: "Is smoking allow?") doesn't work either.
Similarly, "school allows smoking" is a valid sentence. The inverted form would be "Allows school smoking?", but modern English grammar dictates that where there is no auxiliary verb then (unless the main verb is "be" or sometimes "have") we introduce the dummy auxiliary "do" when inverting. So it becomes "does school allow smoking?", as you noted.
"School allows smoking." --> "School does allow smoking." --> "Does school allow smoking?" (First introduce the dummy auxiliary, then invert the auxiliary and the subject.)
"Is school allow smoking?" would be the inverted form of "school is allow smoking", which is impossible, so makes no sense. ("Is the school allowing smoking?" would work, being the inverted form of "the school is allowing smoking".)
