Can I hook up a dryer to the gas line in my kitchen that the stove in using?
You need to check your local codes. I wouldn't be running any hoses for any great distances. I'd extend the line with galvanized steel or black steel piping and tee off to the stove and dryer with shutoff valves. Many locations do not allow homeowners to install or modify their gas lines. Many gas companies will come out and hook up your new appliances for little or no cost and they'll also check the rest of your installation.