The VA is the Veteran's Affairs Hospital... you are basically working for the government. It is difficult to get a job there.
Yes, you can work 3 days and be off 4 but a lot depends on the hospital/floor. When I worked in the hospital, we did a mix of 12s and 8s but the more senior nurses had the preference for the 12 hr shifts. A new grad getting a schedule of 3 12s was unheard of.
The pay depends a lot on your area. Without knowing what part of the country you're in, it's hard to say how the pay will be.
I would say there is no "average" pay check. Your pay check depends on a lot of things... your hourly wage, the number of hours you work, your benefits, taxes, the types of benefits you choose to participate in, how long your pay periods are. If you get paid weekly, your paycheck is obviously less than someone who gets paid biweekly. I'll tell you that when I worked in the hospital, I worked 36 hrs/week. I am single and had to pay for all my own benefits. Some of the nurses who'd been around a while who were only working 24 hrs/week brought home more than I did each paycheck... their hourly wage was significantly higher and they were all on their husbands' insurance.