Call me a nub, but say I'm playing in G major, right?
Say I want to do an arpeggio for the C chord.
Now, the C chord in the G scale would have a sharped 5th, so is there a specific chord name for that?
No it wouldn't; the fifth of C major is G, and in the G major scale G natural occurs. Since it's the IV chord, it would just be standard C major.

G major: G A B C D E F#
No, it would have a sharp 4th.

It's not the chord, it's the scale/mode, which is lydian, the 4th mode of the major key (and C major would be the 4th degree of G major). Lydian is a major scale with a sharp fourth.

If you want to include the 4th in the chord, then it could be called Cmaj#11

...modes and scales are still useless.

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Yeah sorry lol, lil stoned. I meant 4th*. But thanks a ton that answered it, twas buggin' me.