My toilet runs when the kitchen sink is running. I have replaced the fill valve and the flapper and have set everything to the correct heights/levels etc. Also, the toilet doesn't completely run a full flush without using a plunger, which is making me think clog somewhere. I should note a few things, one, the toilet is upstairs, while the sink is downstairs. Two, the toilet is almost 40 years old from the stamp in the tank (flush problem could be jets) and last but not least, we are renters and the house is over 200 years old. We have no idea how old the plumbing actually is. We will be replacing the actual toilet soon, but, I am wondering if the two problems could be due to the same issue and also if there's anything I can do before I give up and call a plumber.
You should call the plumber in. It sounds like you have a failed or failing Pressure Regulator on the plumbing system that needs replacement, this will likely need to be soldered in. Though, you can try tightening/loosening the screw on top of the regulator to see if you can get it fixed temporarily or even permanently...as street pressure may have changed over the years.
As far as the toilet not flushing, a new toilet would be very good. A lot of the time there's a cast iron drain pipe that has rusted itself to a much smaller diameter, which needs to be chiseled and vacuumed out to restore any old or new toilet's proper operation connected to it.