Yeah, you can either Copy or Upgrade. Upgrade automatically updates all components in the design to their latest class. Check it afterwards though, sometimes it won't be using the components you wished for if you have several of the same type( engine, generator, shields, etc.) and tech level. The nice thing about Upgrade is that it automatically obsoletes the last design and lets you modify the updated one( using Edit) before building any of that type of ship/base. Upgrade only works when you actually have new components to use, if you don't then you'll have to use Copy.
You can have both gas and mining components on the same base, but it's not much use since there are no minerals to mine on gas giants, and there is no gas to harvest on planets where you mine. Tech modules can indeed be put on mining bases, just be sure that the 'energy collection' field up on the right is higher than the 'static energy usage' field. Otherwise your base will burn fuel to operate, which is kind of pointless when you can use energy collectors.