Professional liability is synonymous with malpractice. Nursing professional liability insurance commonly available to individual practitioners through such companies as NSO include a benefit for license defense as part of the policy, no separate policy required.
No one knows that you have insurance unless you tell them, or unless an insurance provided lawyer shows up to represent you in a deposition and informs the court of that relationship. Not that it really matters. You are a contract employee of the hospital who carries insurance. They try to shift the insurance liability to the agency who also has a policy covering you. But the bottom line is that they go after deep pockets, and if a nurse makes a mistake, almost always the employer is deemed at fault in an acute care setting. Workload too high, or oversight failure (direct manager, or appropriate assessment of abilities and assignments). It is extrodinarily rare that the nurse involved is dinged and insurance rates reflect that. It does happen of course.
As far as state taxes goes, your work state has first claim on your wages. If you are working temporarily away from your IRS tax home, you will also owe state taxes to your home state (if it has income tax) but you will receive a credit for those work state taxes from your home state. Effectively, the amount of state taxes you pay will equal the higher taxing state. If you are itinerant (without a home) you will owe state income taxes only to the work state for income earned only in that state.
Yes, working away from home in the same general area for more than one year is one way that your tax home will shift to your work state. If that happens, you will lose the tax benefits of working away from home, such as tax free housing and per diems.