Will ATC proactively route traffic around bad weather? When thunderstorms or other severe weather crop up, will the Air Traffic Control in the area proactively route traffic (horizontally or vertically) around the trouble area, or do they leave it up to the individual pilots to request an alternate route?

My intent was asking about commercial aircraft flying IFR, however the information about general aviation and VFR is appreciated
Yes, ATC will reroute flights that are operating on Instrument Flight Rules (IFR).
There are lots of videos of flights getting routed around severe weather floating around on the internet! Atlanta is a great example, as is Memphis, FedEx's hub.
In addition, NASA is working on a product called Dynamic Weather Routes that will help controllers be smarter about amending planned or unplanned weather reroutes.
Sort of, for aircraft operating VFR.
VFR flights that are talking to ATC may get advisories about weather and suggested headings from the controller. I've received notifications about things like 'moderate to heavy precipitation 5-10 miles ahead' many times when operating on VFR flight following.