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I can't find any reasons why you couldn't, but to be safe, you may want to make sure they have just a little room to spread. As long as they aren't planted within a few inches of each other, then it shouldn't cause too much of an issue.

Unfortunately, selective post-emergents will be your best option here. These articles will help:

https://www.gardeningknowhow.com/plant-problems/weeds/using-herbicide-in-gardens.htm

https://www.gardeningknowhow.com/plant-problems/weeds/post-emergent-herbicides.htm

They can take full sun or part shade. Here's an article from the Missouri Botanical Garden all about Vinca major, which grows well in your zone and would suit your purpose: http://www.missouribotanicalgarden.org/PlantFinder/PlantFinderDetails.aspx?kempercode=e212

Our article may give you some ideas, as well: https://www.gardeningknowhow.com/ornamental/groundcover/periwinkle/growing-periwinkle.htm

It appears to be. Either that or a close relative. This isn't so easily treated, unfortunately.

This article will help:

https://www.gardeningknowhow.com/plant-problems/disease/verticillium-wilt-treatment.htm

Do you have a periwinkle (Vinca minor vine) growing outside? It looks similar to that. Mine moved into the corner of our sunroom. There is no need to keep it. I would pull it out.

I believe this to be related to your tomato issue. Phytopthora will infect many genera of plants and trees.