Correlation and filtering
What would the autocorrelation of white noise look like? If I am not mistaken, it should look like a delta function at t=0 since at all other values there is no correlation at all. Is this correct?


This is correct. Of course if you calculate the autocorrelation from samples taken over a non-inifinite time span the mean will be 0 for \$t \ne 0\$, but will be some noise in the output.


  What about when this is added to a signal. Say you have a sine wave and you add white noise. What would happen if you autocorrelate this signal? Would the noise disappear or would it just stay the same or what?


I'm not 100% sure of this, but I believe autocorrelation is a linear process. So you would get an output that is the sum of the autocorrelations of the noise and the sine wave taken individually. This would be a delta at t=0 due to the noise, plus a \$\pi/2\$ shifted sine wave due to the sinusoid. 

Again there would be artifacts if you don't have an inifinite span of samples to calculate from.


  And what if you simply cross-correlated a white noise signal with a sinusoid. Would the correlation always be zero? How is the phase affected?


The cross-correlation would be zero. 

I'm not sure what you mean about the phase. What is the phase of zero?