1. What do you do with a Supply Ship?
Supply Ships provide a one-time amount of manufacturing, not an ongoing increase to manufacturing. Generally, you would build them at a mature core world and send them to a new core world to help them get started. However, I have had the shipyard at a new core world build supply ships that I send right to that new core world. How you use them is up to you, but the main point is that they provide a fixed amount of manufacturing to the core world they are sent to.
2. Do I care about the hex I put an improvement on or only the hexes next to where I put the improvement? (paraphrased)
The simplistic answer is that the improvement will benefit from whatever bonus is associated with the hex where the improvement is built. The more nuanced answer is that improvements not only benefit from whatever bonus is associated with the hex where the improvement is built, but also from bonuses offered by improvements built on neighboring hexes. You really want to plan where you want to build improvements to cluster related improvements (manufacturing, research, etc.) together in the areas of the map that provide the best bonuses. What you want to build will also be influenced by the characteristics of the civilization you choose to play. For example, does your civilization need food? Are they affected by pollution or crime?
P.S. I am just an experienced player offering my advice. I am not employed or compensated in any way by the creators of the game.
