I've heard that strawberries will vine out, but will any breeds grow as a bush so that I don't need to put out lines to keep the berries off the ground?
I think what you want are alpine strawberries. They typically don't send out runners. It's fairly common to grow them from seed, too. Just about any strawberry seeds you fine online will be the alpine-type strawberries. People sometimes prefer these sorts of strawberries when growing strawberries in containers since they don't typically send out runners.
Usually, alpine strawberries have smaller fruits with more flavor and/or sweetness.
Even though they don't usually send out runners, you can divide them in the spring or fall to get more plants.
I don't know if this will solve your problem with putting out lines that you were talking about (it may depend on the variety). As I understand it, you don't want the strawberries to touch that ground. It looks (from an image search) like some go up as in the picture of the first link I shared, and some may be lower down.
I do believe alpine strawberry plants grow taller than regular ones, usually.
NO. And more importantly, expect no more than 2 years crop for any strawberries. (I am squeaking by at 3 years). If you want to ease harvest work get June bearing. If you want all season long berries for your own use get ever-bearing. For a time I worked in strawberry fields with tractor to fert, weed and scrape off. (at the end of one side of the rows was a canal where the front wheels went over the canal and the big wheels pivoted me back...adrenaline). Find new areas to plant strawberries, rotating the crops!
There is a 'Strawberry Tree' but has nothing to do with strawberries. Arbutus unedo. And there is a dwarf variety.