ID of the beer line matters. 3/16" is fine if you have enough. Search for "beer line restriction" here and you'll find the formulas to determine the length you need for your setup.
ID of the gas line does not matter within reason. I'm using 0.170" ID, 1/4" OD gas line, and it works wonderfully. Your gas line is fine. Only difference is during times of high flow (e.g., sealing kegs), I get a little moan from the regulator as it battles line resistance. And I get a lot more flexibility in my lines.
The 5/16" gas line you always see referenced is BevLex 200, mostly because it's so easily identifiable in pictures. It's not any better than plain polyethylene line at the pressures and temperatures we use. Maybe if you keezer served 100 psi and was 150F all day long.