Fun, OK in theory the auto-correlation of white noise is a delta function. But in practice the noise will always have a finite bandwidth, (At high enough frequency this is set by Planks constant, the so called ultra-violet catastrophe of black body radiation, but we rarely get that high with electronics.) and the width of the auto-correlation function is determined by the bandwidth of the noise. Now the sum of a sine wave and noise, is as The Photon said. A delta and a sine.
Cross correlation of a sine and noise will on average go to zero, but on the short term it tells you something about noise at that frequency and phase. (at that time.) (I think about cross correlation as multiplying, so if I multiply a sine by noise what do I get?)
Now here's a fun way to see the auto-correlation function if you have a digital 'scope.
1.) Get a noise signal
2.) Gain it up so it fills the scope display. (but doesn't go off screen)
3.) set the trigger on normal and trigger at the very top of the noise. (The very top is a little hard to define since it's noise, but say so the 'scope triggers several times a second.)
4.) now set the scope to average, with the maximum number of traces. What you see is the auto-correlation function* of the noise. (I can post 'scope shots if there is any interest.)
*OK for the purist's this is not really the auto-correlation function. We call it the qacf (quasi-auto correlation function) and I think others might call it a conditional correlation. (because it looks at the correlation given some condition.. like the 'scope trace was rising and crossed the trigger threshold.) Still, I find this is a powerful 'scope trick for finding the bandwidth of noise. And to see if there is any correlated interference in my signal.
Edit: I just wanted to add that cross correlation is closely related to synchronous, (lockin) detection. Multiplying a noise signal and a sine wave, is a lockin with no signal at the input.