Depends case by case. The VA system is undergoing major policy changes right now. The VA would like to manage 100% of veterans' health if possible so no one falls through the cracks. It depends a bit on your coverage: http://www.va.gov/healthbenefits/cost/insurance.asp as well as the nature of your illnesses. .
Many options. The VA hospitals are designed to treat conditions that have resulted from time in service (service connected). They have expanded this role in many cases and much care is available that is not connected to being in combat, etc. If you have ordinary insurance, you can be treated anywheres and with the Accountable Care Act (Obamacare) you can be treated anywhere unless a new Congress kills this.
Anywhere. Veteran can go anywhere for treatment. Your va insurance may not pay for outside services, but if you are willing to pay for those services.
Treatment options. In addition to VA hospitals, the VA has a number of outpatient clinics, referred to as community based outpatient clinics (CBOCs) . Also, many non-VA providers have the expertise and are interested in treating veterans. If you specifically want to be treated outside of the VA, you should contact your local VA to see if you are eligible for fee based care outside of the system.