third person forms of poder (often puede que, pueda que) often invoke the subjunctive and can be translated with "might" or "may". Alternately, "puede que" can be translated on its own as maybe/perhaps. Though it is not used exactly like that in this case, having "puede" = may does fit in.
Ways of saying maybe: http://spanish.about.com/od/adverbs/a/maybe.htm
May has more than one meaning. May in this sentence means "maybe or possibly" and has nothing to do with permission.
express the possibility or the chances of the occurrence of the main verb: It may rain.
express the willingness of the subject to receive or grant permission or have the opportunity.
I don't believe this sentence is a conditional. It merely a statement of fact (indicative) . A conditional implies some sort of "if" clause.
"Conditional Sentences are also known as Conditional Clauses or If Clauses." https://www.ego4u.com/en/cram-up/grammar/conditional-sentences http://www.englishpage.com/conditional/conditionalintro.html http://grammar.ccc.commnet.edu/grammar/conditional2.htm
If a sentence indicates doubt or merely possible, it could use the subjunctive. -- "Puede que lleva..." (It can be that it might rain.") "Might" indicates subjunctive in English https://www.thoughtco.com/saying-maybe-or-perhaps-3079137
Why isn't will rain correct? I know I'm missing something but I need an explanation why.
The verb after a form of "poder" will always take the dictionary form, just as in English. (E.g. I can wait. He can operate it. They may sleep. The second verb with each uses the dictionary form: to wait; to operate; to sleep rather than sleeps/slept/sleeping.) The only exception I can think of is if the two verbs are part of separate clauses, but then they'd usually have punctuation separating them, as after someone asks "Will you be at the party?" --"I may; I need to check my schedule" ("Puedo; necesito comprobar mi horario"). Hope that helps!
As I understand it clouds are treated as inanimate, in which case using Usted or Tu would be incorrect. However, if you are anthropomorphizing (treating a cloud as a person) then Usted or Tu form would be correct. But I am not a native speaker so don't take this as perfect without research.
I think that would be "Hoy puede llover." The difference is that in the original the stress is on the rain, and in your sentence the stress is on today*. In casual conversation it most likely would not matter, but in others it might. A weather researcher or a farmer deciding when to put up his equipment might wish to be specific.
I take it you're referring to the way an auxiliary (helping) verb such as "poder" (and tener, deber, etc) is always followed by the infinitive (dictionary) form of the verb, which in Spanish ends with -ar, -ir, or -er. English verbs don't have standard infinitive endings the way Spanish verbs do, but we actually follow the same grammatical rule, using an infinitive after an auxiliary verb. We say, for instance, "She can make the cake," "they may go," and "we can ask him"; the main verbs each use the infinitive form "to make," "to go," and "to ask" (instead of makes, made, making, etc).
As for why, the simple answer is "just because"; it's not, perhaps, necessary except that it's standard grammatical usage and therefore the only way that "sounds right." (For instance, native English speakers would know better than to say "She can makes it.") I wouldn't be surprised if some other languages require all verbs in a verb phrase to express the tense, but for English and Spanish, at some point in our linguistic past, this format became common and then became a convention or rule. For a deeper explanation as to why, one could argue that since the auxiliary expresses the tense and mood etc, the main verb doesn't really need to, and that the infinitive form may serve as an additional cue for our brains to help us understand the meaning of a sentence--to perhaps highlight the auxiliary verb and the way it affects the speaker's meaning.
Hope that helps.