Hi:

There are a lot of mulberry species, though the most commonly grown are White mulberry (Morus alba) and Black Mulberry (Morus nigra), as well as Red Mulberry (Morus rubra),  which is native to North America. There are a lot of cultivars and hybrids so size of the trees varies greatly. White mulberry cultivars are readily available in the PNW, so seem to grow well. The plants are self fertile. White mulberry has naturalized in Europe as well as the eastern U.S,. so it grows almost too well in those areas. Which to choose might partly depend on when you want the fruit: white and red mulberry fruit in spring while black mulberry fruits in fall. The black mulberry fruits are especially soft and tend to collapse when picked. All of these plants respond well to pruning so you might have a lot of flexibility in maintaining size. They naturally form trees, but simply pruning the main stem would create a shrub. I'm not aware of any information on differences in nutritional content. The berries are very perishable though, so need to be processed quickly. Feel free to write with additional questions.