Hi Alison,
Hmm .. so shoots coming out at the base looks like it's coming from the graft which is a whole different bush. Roses are grafted on to really strong root stock that is resistant to disease and such and usually doesn't produce blooms or if they do they are totally different from what you bought.
When you go to cut the canes, remember use a 45° degree angle and try to cut just above leaf nodes that face away from the center if you can identify them. After cutting, I'd spray the cuts with a good fungicide. If the canes are still viable you should see new growth in a week or so if you nights are in the 60's and above.