1. depend on the traits of the ambassador. When he give a good trade income bonus, you should assign him to close/big empire to raise your income.
2. You should see the max. fuel range as white circle. When you don't see any cicle at the Galaxy view, the range is maybe larger then the view.
You can setup a refuel threshold for fleets at the empire policy when auto. fleets should go for refuel. Manual fleets you need to assign them self to refuel.
3. Resupply ships are a kind of Gas Mining base with engines and Hyperdrive, you can use them as forward refuel point. You can deploy the resupply ship at any Gas cloud/ gas giant with Caslon then the ship start to harvest Gas and other ships can dock at the resupply ship to get refueled.
Got the Resupply ship enough fuel gas at the cargo, you can deploy it somewhere else too, closer to the point your fleet are.
4. You can see how many troops are on a planet when you got a state ship at the system. The exact strenght you only see when you attack the planet.
5. You should allways see them at the planet facility . If you mean how to destroy them. Open the planet screen (double click at the planet name or select the planet from the colonies list), then under facilities you should see the hidden base, select it then click at attack. But you should have some ground troops ready there.
6. Thats the strategy question ! :-) After you played a few games and designed your own ships you get the feeling about the techs.
Weapon: you should concentrate at one weapon, it doesn't matter much what kind. But Gravity weapons arn't effectiv to kill ships, but very good for defence base/station they damage ships and let them return home for repair.