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I have been studying the score of an overture by Mozart in MuseScore and I noticed that the woodwind instruments and the one of the violins were playing the supertonic chord with the 6th degree included, and the other instruments were playing the dominant chord. How does the composer decide which chords grouped in which instruments will sound good? Is it only of the composer's whims? Also, can the different chords get discordant?

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That would make the chord sound like a dominant 9th. In C, for example, the supertonic with a 6 would be spelled D,F,A and B. The dominant would be G,B,D and F. There's only one different note in each group, and blended together it all makes G9. A fairly common set of notes.

Yes, a composer can and does split notes in that way.In an orchestral setting, there is some spaciality which can make notes which may sound dissonant sound o.k. together. Of course there's also the difference in timbre and volume between instruments, which also helps.

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You mean the harmony was a dominant 9th but the 9th only appeared in the woodwind? We have to wait a little time after Mozart for true polytonality (and even then, the combination of two chords can often be analysed as an extended version of one basic chord).

Are you looking at "Marriage of Figaro" overture? Which bar? The constant use of pedal points is a striking feature of that piece.

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