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North & Hillard Ex. 213; Q5: the following is to be translated into Latin: "I am willing to send anyone at all to find out what is going on."

The answer: "volo quemvis (quemlibet) mittere cognitum quid agatur."

A few points: firstly, two-words-joined-into one, "quem vis" whom-you-want & "quem libet" whom-it-is-pleasing; but doesn't "libet" require dative + infinitive construction giving "cuilibet mittere"?

Secondly, what does "cognitum" mean here; it cannot be "having-been-learned" when the intention is in the future? Is it an acquaintance--the person being sent; if so, how is the concept of "finding out" to be expressed?

Thirdly, present subjunctive passive "agatur", after "quid", giving (literally) "what-may-be-done" referring to the future; but the Q asks for present tense--"what is going on".

On quemlibet: it's not the "whoever" that is being pleased, so the quis/quem isn't what goes into the dative. I suppose there's an understood mihi, or tibi; but I suspect that really, libet has become lexicalised as a suffix, separated from its verbal root.

On cognitum: I think it is a supine, not a participle. I don't remember seeing mittere cognitum before - I'm only familiar with this use with ad: "ad cognitum" - but I think that's what it must be.

On agatur: there isn't a future subjunctive anyway, but why do you think this is has future meaning? The meaning is what may be happening (being done).

1. 
   

   A couple of indefinite pronouns and adjectives consist of a verbal element + quis/quid or qui/quae/quod: quivis, quilibet, and nescio quis. These are tricky because the verbal element doesn't affect the form of quis or qui. Instead, that form is determined by the function of the overall pronoun/adjective in the clause.

   
   
   

   If the pronoun is the direct object, the accusative case is needed, as always, and the form would be quemvis, quemlibet, or nescio quem. To show possession by the pronoun, the form would be cuiusvis, cuiuslibet, or nescio cuius. To show that the pronoun is the indirect object, the form would be cuivis, cuilibet, or nescio cui. And so on. The verbal element doesn't really function as a verb in these words/phrases, though it does color the meaning of the overall pronoun.

   
   



2. Cognitum is indeed a supine, because when you send someone to do something, some degree of motion on that person's part is involved. You could just as well use a different construction here, such as ad cognoscendum/cognoscendi causa (gerundive), ut cognoscat (purpose clause), or qui cognoscat (relative clause of purpose).



3. 
   

   The present subjunctive in a subordinate clause (here, an indirect question) may refer to action that's later than the action of the main verb, as you say; but it can also refer to contemporaneous action. In your sentence, the subject is sending someone now to learn what's going on now.

   
   
   

   Also, as a general rule, subjunctives in Latin indirect questions should be translated without 'may,' 'would,' etc., because they're reflecting questions about actual facts. Obviously, there are exceptions; but this is easily one of the most common corrections that I make with my students.

   
   
   

   Therefore, cognitum quid agatur means 'to learn what is being done/carried out/performed/etc.' or, more colloquially, 'to learn what is going on.'