Can I run THHN for separate washer and dryer circuits through the same conduit?
It is permissible to run more than one circuit in a conduit as long as the conduit is sized to carry all of the current carrying conductors. In your case a 1/2" conduit should do it, but I would go with a 3/4 for ease of pulling.


2(a). One of the advantages of pulling in one conduit is that you would only need a single grounding conductor sized to the largest circuit.The ground is attached to the receptacles and any metal enclosures and there are places marked for that purpose. You will attach the ground to the neutral bus in the main panel. You might check to see if there is a proper ground in the panel and make any necessary upgrades.

2(b). Most jurisdictions require any work to be permitted and installed by qualified personnel. We really can't be certain of every jurisdiction since codes vary by state and municipality. A quick call to your AHJ can solve that problem.

Good Luck.