A1:  Mining star bases are used to extract resources from the special nodes you can see floating in space.  You then spend these accumulated resources to build special buildings, improvements, and ship components.  You can also trade them, but I don’t recommend it most of the time as other nations don’t offer you enough in return.  I do buy from others when I am desperate.  The engineering tech page has a path which will allow you to slowly upgrade your mining star bases and ratchet up the income stream.

 A2:  Distance from the home world has no impact on mining starbase income.  Whatever is covered by the starbase area of control and not already claimed by someone else can be mined by you.  Normal AOC is radius five.  Vigilant trait in race creation ups this to radius six.  There is a path under the Pragmatic Ideology which gives +2.  Very deep in the tech tree you can raise this again.

You will always need Durantium and Prometheon, try to prioritize those ASAP.  Hope this helps!

BTW, is that Jewel Staite in your avatar pic?

If the mining base includes a precursor item (the diamonds), you can add a module to "harvest" that as well as any special resources.

If you have the correct government types, the market tab becomes available and you can purchase the special resources for a fee. Of course, if you are mining them, you can also sell them there.

The AI's are much better now in building bases to obtain these, so building advanced scout is more critical if you want these. The production is much lower since Intrigue, so each mining base is more critical to success.