The verb is आना (to come) and it is in the present continuous tense for a male subject (आ रहा है ).

You can see that in a similar sentence- आज वह खेलने आ रहा है। (He is coming to play today).

The way to say nobody in Hindi is कोई नहीं (literally, 'not somebody' or 'not anybody'). So, the sentence 'No one is coming to play today' would be 'आज कोई खेलने नहीं आ रहा है।

However, in present tense negative sentences with 'नहीं', the है is optional. So, आज कोई खेलने नहीं आ रहा। is also right.

Yes, because you would be separating नहीं from the main verb in the sentence (आना). "Aaj khelne koi nahin aa raha hai" would be acceptable though.