Space stations: What are the point of these? What can you do with them? It seems that they only serve as extra planetary defense and maybe an extra ship yard?

Weapon platforms: I have no idea what these are for... doesn't seem to be possible to study them in the simulator.

Satellites: I've been able to add them into the simulator and they do as I suspected. So as long as I have them in the planets cargo, they're in orbit and will automatically defend a planet, or do I have to launch them into orbit somehow?

Space Stations are great for defending important locations like Space Yard planets.
You can set them in destroyed systems with a Space Yard component and Fuel for closer support to the front. Basically they're improvised bases/strongpoints if no good planets are available.
Something I often use them as is "Forts" on warp points where the AI like to send ships and small raiding parties now and then. I hate having to hunt them down so a stationary base with lots of weaponry and defence stops most of the raiders. Of course you can also use satellites and mines for this. But space stations is more fun. :)

Weapon Platforms are great and relatively cheap planet defences and so are Satellites. In my border systems I make at least one weapons platform, launch a bunch of satellites and make some troops on every planet. This makes for a great in depth defence so if you are attacked you usually have time to get your fleet there before the whole system is lost. Missiles are specially effective on weapons platforms.
I've always launched satellites as soon as I've built them so I don't know if you really have to do that. Launching them frees space which is a plus on small planets.

Yeah. Just leave the weapons platform in the planets cargo space.

I almost never use satellites except on planets. Build them there and then launch them when they are built.
But if you you want them on say a warp point you use a ship with a satellite bay to get them from a planet to there and launch them. Or you can use a ship with a space yard component and a satellite launch bay to build and launch them on the spot you want them. It's very practical to have a couple space yard ships for building things here and there.

The reason I'm not fond of using mines and satellites on other places than planet defence is that they need to be replaced. A space station is much more expensive but more durable and if built right can last a long time before it's destroyed or in need of repairs.
The AI loves placing mines and satellites though. I can't be arsed to do that everywhere. :P

Is it possible to use the ministers to automatically use population transports to spread out my population between colonies?

My memory is foggy about this but I think you can mark what kind of things the minister can take care of and then mark a ship/fleet/planet as minister governed. So yes I think so.

Edit: Try this.
Empire Status -> Ministers -> (Check.) Transports
Create a Population Transport, click it and then click Toggle Minister Control.

Thanks!

I actually tried that before posting and it worked for one trip, then the ship just sat there doing nothing at all. Oh well...

Well the ministers are a bit flakey. :) That's why I don't use them much.
I think I remember having them to do just what you're trying to do and getting it to work. Keep trying, it might be something small that you have missed like putting them in the Population Transport category when creating the design or something else.

Edit: Have you set your ministers to automatically resupply? I think that's what it does. Or something. Another thing to try if you haven't researched that endless supply thing yet and have it on your ships.
There are lots of small things everywhere in SE4 that are interconnected so if the chain is broken somewhere it can be difficult to find what's causing it to not work.

Is it possible to take an old, outdated ship and upgrade it?

Also, I'm still sketchy on what the 'retrofit' option does.

That is what the Retrofit command do once you have updated the schematics for that ship. :)