I’ve been playing the demo version of GSB and I’ve been waiting until I’m able to buy it.
Problem is, I plan to upgrade to a new computer in the near future. Would all my paid features (and more importantly, the support for them) be carried over to my new computer? I don’t know when I’m buying, so I need to know if everything would still work or if it’s better to wait.
If you were to purchase GSB today from Positech direct you will get the install files and play today.
(well as soon as your purchase is processed anyways)
When you migrate to the new computer, use the same files / code to reinstall on the new machine.
(Just remember to migrate your GSB directory located in MyComputer to transfer your progress / honour etc.)
But yes, everything should still work - only faster. Plus you can play some of the larger mods that require DLC’s
So, you get a code for downloading? Which means any future updates can be downloaded without issue?
Firstly: the local Forum Sherriff says welcome aboard, Tek. Enjoy your stay with us.
The GSB forums are thinly populated these days but we do also have a very good signal-to-noise ratio, and we strive to keep it that way. If you have additional questions, feel free to ask them.
Most of the forum activity these days is centered around the Gratuitous Modding subforum; drop by and take a good look around. 
Oh, and fellow noob Redd13 has a very instructive thread going on right here which should be of considerable interest to all new players: Diary of a Space Tyrant.
You got it.
You can always re-use your original GSB download link over and over. If you buy the game directly from Positech (recommended!), the email which includes your special info is sent to you by BMT Micro. And if that link should ever become broken, cliffski (Cliff Harris: GSB’s developer and all-around cool dude) has repeatedly proven he’s more than happy to send you a new link.
Regarding your second point, technically what happens is that once you manually register your copy of GSB via your username & password, your game install will check-in with the Positech GSB server once every 24 hours. Should there be a patch available, it will be downloaded and installed automatically. This late in the game’s development cycle, patches are very rare but there is a new & slightly buggy one out there that’s just a few weeks old and which, itself, is possibly going to be repaired via a subsequent patch Real Soon Now™. 
Thank you for the welcome. I’ve got a few mods planned, and I hope to start getting into that soon.