Can shopping be an unhealthy addiction?

It could. Shopping for women is a common way to deal with stress. The problem is the excessive spending when people could not spend more money that what they can afford or when they accumulate things and have difficulty to get rid of them. Addiction means the need for more and more and signs of withdrawal when stopping it, ang give priority to the addiction versus responds abilities at work or home.
It can be. Shopping addiction is the subject of new research. "behavioral addictions" altogether (e.g., gambling, pornography, sex...) is a new and subject & remains the topic of some debate. But imo it is clear that behavioral addictions, including shopping can become severely damaging to people's lives. In my experience it is best dealt with by an addiction expert. Ask your md.
Yes. It is a form of anxiety and needs further help if it causing serious financial issues.