Hi, 

The Spice bush sound like that is a nice addition! The deer will probably continue to nibble on the tender young branches but once the plants get big enough, they should hopefully leave them alone.

Now for the hard news: 

The vinca is hard to eliminate since any piece of the stem or root with produce new foliage. You can visit the attached link to see different control methods. You can also use a glyphosate material on it to help kill the root system. It may take a few applications in a row and raking up any foliage to help see any new shoots that come up. You will probably need to use the glyphosate in conjunction with a "surfactant" or "adjuvant" to help it stick and stay on the leaves since Vinca has that waxy coating. It will have its own instructions for dosing on the label, and is only combined with the herbicide upon mixing-up a batch to apply (otherwise they are stored separately). You will need to check the herbicide label about the ability to use a spreader-sticker though (the surfactant) since some formulations might already include one or might be incompatible with one. 

For the Rose of Sharon you can remove the main branches and cut down, then make some marks in the stump and spray/paint/dab on the glyphosate (or triclopyr would probably work too, but glyphosate tends to be a little more 'safe' and doesn't last in the soil). You should not try to spray the upper branches since this could cause the herbicide to drift and damage other plants near by. You can apply to the stump and any foliage that re-emerges after you cut it down. 

Let us know if you have more questions with this. 

Emily