Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't? Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't?

Yes.  For example take $C$ as a Cauchy random variable and independently $H$ as $0$ or $1$ with equal probability.  
Let $X=CH$ and $Y=C(1-H)$.  
Then the expectations of $X$ and $Y$ would be half the expectation of $C$, except that it does not exist, while $XY=0$ and so $E[XY]=0$.  
