Yes, they will freeze over the winter. The earth provides an amazing amount of warmth. Since the plants will be exposed to cold air on all sides, they will freeze out and die. You could surround them on all sides with bales of straw (under the wagon, too), but that might not be enough. Another option is to dig them up and put them in a garden bed for the winter. You would want to cover them with straw after the ground has frozen. That keeps the roots from being destroyed by freezing and thawing the ground. 

The last option is to treat them like annuals. You would have to re-plant every year, but the wagon would be really pretty. Many commercial strawberry growers in the south only grow their plants for one season and then replace them. We normally recommend removing the flowers from first-year plants so the plants can concentrate on developing a good root system. If you use the wagon, I would let every flower grow into a berry. 

Not really. You should probably stick with annuals. A raised bed like you have would need perennials hardy to maybe zone 2 and they are hard to find. There are some self-seeding annuals or annuals that are easy to start from purchased seeds every year that will keep the cost down. Another option is to grow perennials in containers and move the containers to a sheltered location (or cover them with straw or shredded leaves) every fall.