I don't know. All I can related is my personal experience, which is: I've dug up and transplanted blackberries, black raspberries aka "blackcaps", and dewberries from the (now former, McMansion-covered) field behind my property and planted them WITH my cultivated brambles, they've co-existed for many years, and everyone gets moved around and replanted as beds come and then wane. No problems with disease. I kind of regretted the dewberries, they're not very productive in terms of fruit output and they're very invasive and painfully thorny. As they say, your mileage may vary.

It isn't that simple. There are distinct differences in how one prunes a black raspberry plant to a red raspberry and also differences in how one prunes a red raspberry related to whether it's a summer bearing or an everbearing, and how and when you want fruit. I'm enclosing a link you might want to read on pruning and maintaining raspberry brambles. Here is a link that might be useful: raspberry

Konrad is right about the Round-Up or other brands of glyphosate. They take a little while to die... but they do die. If you are concerned about not killing the lawn too, go with 2-4 D. Wild raspsberries can be tough, but usually 2 sprayings will get the job done even with 2-4 D.

If you separate the bulbs, make sure you have a minimum of three bulbs for each division. Five or more is better. Any less and you can set the plant back for years. Make sure the new pots will just accommodate the roots and a bit more. Too large a pot can contain too much medium that will stay wet without enough roots to utilize the mosture, and the roots can rot.