Do dishwashers and clothes washers require dedicated circuits in an older home?
I do not believe either of these situations is hazardous.

However, keep in mind that codes have changed over the years to require a lot MORE circuits in the kitchen than (typically) there used to be. In particular, you should have at least one circuit for small appliances, separate from the disposal, dishwasher and other built-in appliances. But if everything is installed correctly then it is perfectly safe as is, with the catch that if you run a toaster AND disposal AND the dishwasher is doing a heated dry cycle (the motor &amp; controls the rest of the time use relatively little current) then you may trip the breaker.

A radon fan should not use much electricity - I wouldn't be concerned about combining it with your washer circuit.