You can do all this by using the Direct Selection Tool (the white arrow) or the Pen tools.
Right--but, is there an easier, automated way to do this?
Additionally, can multiple path points be grabbed with the white arrow? (I have tried and can only seem to grab one-at-a-time, which makes the process very cumbersome and time-consuming).
Hold the shift key down and select away each anchor point you select will be added to the selection be aware if you select an anchor point twice with a double click you deselect that anchor point.
You can try to simplify the path as well Object>path>simplify
You can also try to develop a better more responsive setting in the Live trace options but that requires some experimentation.
Automation plus complete control simply doesn't happen. You must choose one or the other or various stages of both. Sounds like you've used all the automation you can and now you need to step away from it to get the complete control you want. Illustrator can't read your mind about where you want an anchor moved or a path thickened.
There's more to it than clicking "Trace" and saying "voila." Try experimenting with the Trace options dialog box. Change the Corner Angle to 0 for starters. This will give you a base of straighter lines.
After the the trace is expanded, go to Object>Path>Simplify and choose "Straight lines. See how that looks.
Lastly, if you're not too keen on the Pen tools, go over the vector lines with the Smooth tool (under the Pencil).