120v house equipment with 240v generator question
To answer your specific question, the difference between L14-20 and L14-30 is maximum amperage, not voltage. Both carry both 120 volts and 240 volts (actually two legs of 120 which can be combined for 240).

As you suspected, the combination of the adapter and cord are useless (as opposed to a direct L14-30 to L14-30 cable). Actually, it’s worse that that. The L14-20 connectors are only rated for 20 amps. Since the generator can produce 30 amps, those connectors could melt or catch fire!

I’m assuming your father went shopping for a 30 amp cable and the sales critter sold his those, what it had, making up the story about voltages. 

For safety, you really need an L14-30P to L14-30R cable. Since you don’t have one, I suggest you follow @JACK’s advice and plug the appliances you need directly into the generator instead.