I'm still a gardening neophyte, so I don't have a lot of suggestions. But I LOVE the potential of this garden. I think it could be just gorgeous with the right kind of plants and shrubs. One thing I can comment on, is that no matter what you plant, I would get rid of that border pronto! It does NOTHING to contribute to the look of the garden, other than differentiating it from the lawn. Rip that sucker up, and replace with some pretty stones or bricks of some kind. That alone would make the front garden look more attractive once you've gotten rid of the weeds.
So... Under all the weeds and grass is old mulch. can I still do the box method? or Does the mulch need to come up?
Mrs.Eyring---You might take a look at the soil underneath the mulch. If it is very poor or mainly sand, you might want to add a layer of compost on top before you put the cardboard down. Just lay it on top of the weeds, then the cardboard, then the new mulch. My other thought is that you may want to find out what those shrubs are before you just yank them out. Maybe they are something worth keeping or worth transplanting elsewhere. Maybe they bloom in the spring, for example. A garden center should be able to identify them for you if you take a branch in. Good luck!
Cayuga's suggestion is a very good one. If you do take a branch into a gardening center, make sure it is enclosed in a sealed clear plastic bag. Just in case, if there is any disease/bug involved, garden centers are very sensitive about that!