What to do?

What to do?

So I love roses, and I've been quite happy with how hardy and resilient they've been for me. However, I've encountered an issue, that I'm not sure what to do with. I was away for almost two years, and the roses didn't get the maintenance they needed while I was gone. I have...SUCKERS. Which, normally, I would just dig up, and that would be that, but they've been growing so long, they've actually started to bloom. Apparently, the Floribunda I have was grafted on Hybrid Tea stock. So, I have two different color roses, with different foliage, growing off the the same root system. Can I leave them? Should I take up the suckers? I'm torn. What is best for my rose in the long term? The new flowers are already blooming The original flower (which is a 4" bloom in a pretty peach) is about to bloom And here's a picture of the "root" of the problem! I need your help. My Sunset Western Garden guide doesn't cover this one!
plant id please

plant id please

Need ID on lawn denizen…

Need ID on lawn denizen…

Melons

Melons

Plant ID help please

Plant ID help please

The cabbage family 2009

The cabbage family 2009

Yet another plant ID.

Yet another plant ID.