1. Commercial shipping will automatically handle trade goods. And on command you can set up a contract (civilians/ind status tab) to have them move structures and installations. They'll also move colonists, and harvest from gas giants (assuming you have the necessary techs)
They can not move ship components, PDC components, missiles, maintenance, or minerals.
2. Trade goods are constantly generated on all planets with people on them. Also they always generate a demand. This can be viewed in the Wealth/Trade tab. If there is nothign present below, its because you haven't passed any time yet.
All colonies with a colony cost demand infrastructure.
Once the planet reaches a few million, it'll open up a handful of basic goods demands, (machinery, construction materials and such)
And i believe when it hits 10 million it opens up all of the goods.
Each planet will generate infrastructure, and a random spattering of the trade goods (never all of them) and this rises as the population gets larger. The demand for the things they lack also rises.
Say you colonized the 4 moons of Jupiter. After they hit 10-20mil pop, they'll start trading between each other slowly, and generating extra wealth for you through tax. Once they reach 100mil pop, they'd start making you a lot of money. But of course you do need to limit your civilian shipping at times due to processor limitations (can have literally thousands of the things running around and it'll slow down your game)
3. They can generate a trade run with a colony within 4 jumps of their starting planet, as long as there are gates on the jump points as needed. (means the civilians don't need to have jump drives). They can travel say 20 away, as long as they had 5 different trade runs between planets on the way. Though trade ships tend to favor your local systems better than far out ones.
4. I believe that there is more money generated from an empire>empire trade, than generated by just trading between your own planets. Also trading can generate more goodwill between you and another empire, making alliances easier.