Well, hard to say what your issues are. But yes, a clogged cat can cause a car to not run at all or run bad. If your cat has never been changed, I would say just do it. These things only last about 120k, more or less. They aren’t expensive; just have a local muffler shop weld in a universal cat. It is much cheaper than buying a whole new assembly.
The potential electrical problem, I guess the battery is a good place to start. I’d have your alternator checked also. But who knows, you might just have just dowsed something in the rain.