Is the construction 'Let + subject + verb' considered as an order/imperative for the third person:
Let every man count his days
when it is intended to mean 'must'/'is ordered to'?
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"Let" can introduce an imperative phrase, but it doesn't have to. Third person is a red herring here; "Let us go forth in peace" at the end of Anglican communions has exactly the same force of command. Exactly how strong an imperative it is depends very much on the context, though. It's not terribly strong in either your example or mine, but "Let my people go!" is a different kettle of fish entirely.
If I recall correctly, "Let + verb" is the standard way of translating subjunctive "imperatives" from Latin.
"Let" can be used in the imperative mood for the following purposes:
A command insisting someone allow something to happen
Let him go!
A polite way of making or responding to a suggestion
Shall we go? ... Yes, let's.
A polite way of offering help
Here, let me.
Expressing a strong desire for something to happen or be the case
Let him be all right.
Expressing defiance or a challenge
If he wants to walk out, well, let him!
Expressing an assumption upon which a theory or calculation is based
Let A and B stand for X and Y, respectively
These are all in imperatives, though only the first would actually be an order/command (and maybe the fifth). The fourth would be the one you're looking for, and it is in the imperative mood (though not necessarily an order/command).
Source: New Oxford American Dictionary
Unless it is followed by some indication of consequence or ultimatum, I would not understand it to mean "must" or "is ordered to." For example
Let every man count his days, lest he become too proud.
Let every man count his days, for every day is a treasure.
In the first example, it sounds as though every man must count his days, because some consequence (becoming too proud) will fall him if he doesn't. In the second, it is not an order or imperative, because there is no consequence.
I think that that construction does form an imperative, but it is more lax than a traditional construction. For example, consider:
Let us rejoice!
In the former case, I wouldn't be worried if I decided to refrain from festivities. But in the latter case, I might be a little concerned.
I see a subtle difference between the two, but I don't know if that goes for everyone.
Grammatically, I think Let every man count his days is the same whether it is intended in the allow sense or the exhorting sense. It is grammatically like Make every man count his days or Help every man count his days. Of course the meaning is very different.
The main clause is imperative, and as usual with imperative clauses, its subject is omitted: Let/Make/Help [direct object] [bare infinitive].
So to answer your question, I wouldn't call this “an imperative for the third person” in any grammatical sense. Semantically, perhaps. Grammatically, that would mean the subject was third-person, and here there is no subject.
However, some imperative clauses do have subjects, and the subject can be third-person:
You stay put until the ambulance comes. (second-person subject)
Heaven help us all. (third-person subject)
Everybody chill! (third-person subject)
You could almost interpret these as ordinary subjectless imperatives with a direct-address tacked on the front, but in that case, I would expect the direct-address comma to be almost mandatory, and instead it makes all these sentences much worse: You, stay put. Heaven, help us all. The Cambridge Grammar of the English Language astutely points out that Nobody move is grammatical while Nobody, move is nonsense.