The way the Spin Control Center works is, it makes any ship in orbit around the planet on which it built appear much stronger. If you put your ten strongest ships in orbit around that planet, your military rating can increase a lot. This can keep a strong, hostile AI off your back.

Yes, it's a Super Project. You can only build it once, so it only affects one planet. The only Super Projects that affect your entire empire are the Hyperion Shrinker and Hyperion Logistics System. However, all Trade Goods and most Galactic Achievements affect your entire empire.

Well in a way it does By improving the 10 ships by a huge amount your entire empire looks stronger in the eyes of the AI's, thus stopping them from being outwardly hostile towards you.

Many players use 'Paper Tigers' with the SCC as well so they dont get bogged down with hugely expensive ships sitting in orbit doing nothing. By 'Paper Tiger' i mean a cargo hull with only 1 point of structure (hence fairly cheap) with your best weapons strapped onto it. The ship then has a huge attack rating but would last a single shot in a fight, so you are only creating it to look good with the SCC.

Hope that helps.

I imagine that having a military starbase with ship-aiding modules will stack with that?

It's worth noting that every player can build an SCC, so it's possible to have more than one in the game if you capture one, after having already built your own (works just like shrinkers or manu/econ capitals). I've had three or four before in a single game.

Don't see why that wouldn't work. Military starbase ship assist modules do enhance ships in orbit and should stack with the SCC. Could be worthwhile if a planet is located in the right place and enough military starbases can be built around it. Though, you're talking a good number of constructors. Might be better off applying those resources to better ships for the SCC.