Hello, and thanks for your questions!

Here in Minnesota, we have many White Mulberry trees, and YES sometimes a tree will only have male flowers and not produce fruit.

Even though the berries might be dark, they are still most likely a white mulberry tree since red mulberries are very, very rare here. Ripe mulberries are edible, and birds love them whether they are ripe or not. Too many mulberries can cause some indigestion. If you submit some pictures of the trees and fruit, it could help to make sure you're not eating something else (it's never a good idea to eat unknown berries).

Mulberries produce fruit only in June or July, and yes they are a messy tree.

Here are a few links to mulberry tree information:
This site is by the University of Minnesota urban forestry department. It has some helpful pictures https://treetrek.weebly.com/white-mulberry.html

This is a link to the UMN Extension site with a little info on mulberry trees, but it doesn't have pictures: https://www.extension.umn.edu/garden/yard-garden/trees-shrubs/mulberry/

Here is an article about how RARE red mulberries are, and that is why I think you most likely have a white mulberry: https://www.dnr.state.mn.us/mcvmagazine/issues/2017/may-jun/red-mulberry.html

Lastly, the Missouri Botanical Garden has a page that has a little more info on these trees: http://www.missouribotanicalgarden.org/PlantFinder/PlantFinderDetails.aspx?kempercode=d439