I think it's fairly common at most programs that first year moonlighting is very restricted or not allowed as most programs front load all the call and busy service obligations to first year and since moonlighting counts towards hours (in house or not) programs are not interested in messing with duty hours. Just and FYI. Lots of fellows moonlight, however. It helps if you have your boards passed too.
As far as the specifics this seems to be very market dependent and it usually also helps if there is a VA. I am lucky enough to moonlight in a market with a lot of competition for moonlighters and I'm credentialed in a few spots so the pay is nice. At one place I only hold a code pager and respond to RRT or codes and get $90/hour for it. Other places pay around 150-170/hour to cross-cover and $100 per admit at night. The ED (at a VA) pays the best at $200/hour. My malpractice (with tail) is covered by my employment at these places as they all use "employed" moonlighters (and the VA doesn't require malpractice). This is also nice because they take out federal withholding this way. I don't think I'd work 1099 jobs as a fellow that required me to come up with my own malpractice.