1. Is there a way to automate the design of certain types of ships? My goal is to create my own military ships, and let the AI design mining ships, passenger ships, etc. Is this possible?
Go to the Empire Policies screen and set ship design to be fully automated. Then, either look through the checkboxes in that section of the policies screen and remove the checks from the boxes for the ships that you want to design yourself (and add checks to any boxes for roles that you don't want to design) or go into the ship design screen, go to the column titled "upgrade," and click the words in that column until it says "manual" for the categories that you want to take care of and "automatic" for the categories that you do not want to design yourself.
Be aware that, at least according to the tool-tip, ship design must be set to automated for the computer to even think about creating its own designs; if ship design is set to manual, the game's help says that the computer will not create designs even in roles with automated upgrades. I've never really checked to see if that's true or not, but it's something to be aware of if you want to have control over the military ship designs but not the civilian designs.
2. How do you upgrade a small space station to a larger one? I can't figure this out.
It depends. If you're talking about upgrading a spaceport from, say, small to medium, then select the spaceport, right-click somewhere nearby, find "retrofit" in the menu that pops up, and select the class of spaceport that you want this spaceport to become. If you don't see the option, try retrofitting to the latest design for its current type and see if that fixes it. Also make certain you have enough money to perform the retrofit (the listed cash on hand in the upper right corner is not kept fully up to date; rather, it updates periodically but may be inaccurate between updates, though you can now force an update by opening a menu like the ship designs list) and that the station is not currently undergoing an upgrade. If you're talking about upgrading a non-spaceport space station, then you have to have a larger design with the same role in order to order the station to retrofit to a larger design.