When i flush my toilet, the water goes out of the toilet, but then it goes silent (doesn't fill back up) for a minute or two. I can hear the water moving through the pipes right after the flush. Then after a minute or two, the toilet kicks on and starts filling the bowl again. Also, sometimes I get a sewer smell in my bathroom.
if it's a Fluidmaster fill valve, it sounds like the rubber valve seal needs replacing. Only a couple of bucks at lowes/HD or any hardware store. Takes about five minutes, and instructions are on the back of the package. Sewer smell is likely a different problem.
check and see if the float in the tank is stuck against anything. i see a lot of wc's like that where the ballcock is jammed against the back of the tank, and it takes it a minute to fall down and turn on.