Just watched the video, and it looks great. Very nice concept there. Too bad with hangars particularly, but i wouldent want every faction to have the same fighters, nor cargo ship so same there. Anyway its a minor annoyance compared to this cool stuff.
Abilities: you could also let Max Allegiance be a race specific uppgrade for someone suitable, that would leave one more slot for other stuff.
-------I'm also thinking Orbital Defense should be an upgrade that combines weapons and health together to free up some slots.--------You could do this, or perhaps it would be better to combine the health and shield into orbital defence/fortification/whatever instead. Maybe a bit more synergy with both beeing defensive in nature and a natural symbiotic project. Dont really matter offcource its mostly flavour.
Dooing these things would let the general abilities be: population, culture, factory, defense, offense, block colonice, that would leave 3 abilities for race specific ones. Gonne need some ideas there though. 
Can always steal some from the inividual starbases for the easy solution or go all out creative and tailor 2-3 for each faction.
Stock market could be one for hypercorp, but it could just aswel be a general research in the research meny, or a special building that substitutes for regular trade buildings, instead of trade ships you would have "Stock Traders" lol, maybe that would be too silly, hehe.
By the way, have you tested how it looks when weapons fire are coming from the moon? I am a little curious as to how that will look seeing the natural shape of the moon. Also i assume that different factions will get different weaponry, same as a starbase?
Hypercorp: Stock Market, Cloning facility, Hypervelocity Cannon: JUST BECAUSE I WANT THEM TO HAVE SOME DPS!! (also it matches hypercrop in the hyper part) 
Rogues: Tortuga:The moon functions as a focal point for all sorts of piratish activity making the pirates more effective at...beeing piraty, all their ships gain increased movement speed in the local grav well, and you get a slight economical increase. You get the idea. Historical name ofc.
Corrupting influence: Beeing so close to wealthy and corrupt Tortuga can make even the most stalwart man think thoughts of greed: Capture enemy frigate, long cooldown. This could be the followup research to the number 1: Tortuga--->Corrupting Influence
Nephilim: High temple/Visions of heaven/Joyful Reunion/etc: The Nephilim can summon forth blissfull visions of heaven that makes those exposed to them unable to fight for a duration. (the standard abbility that makes all ships ceasefire)
Cathedral/Healing touch/Gods Love/Love of the Creator: The Nephilim can call forth waves of healing that affects all nearby beeings.(ships haha) (a variation of repair)
Eternal Peace: The Nephilim all prays for the day when everyone shall join them in eternal paradise(increase culture)
Eternal Bliss: The Crew of an enemy vessel is exposed to the true light and immediatly switches sides, permanently.
Plague: Biological Research Facility: Creates a deadly plague to wipe out a specific population. Maybe a little overpowered but could work like a small superweapon hehe. Target a planet and let loose a nasty disease that lasts for a long time and kills off most the population and hinders their regrowth for a long time. Would need to be balanced with respawn rates so as to not be op. Muahaha.
Spore Factory/chimney: Saturates nearbye space with deadly spores that attack even the very hull of non-plague vessels. In effect let every enemy ship in the grav well take damage ALL THE TIME. muahah. Would be similiar to the pirate cap ship abbility of the AnnaV, but would need to be lower ofc, maybe 0,5 hp first stage, then 1, then 1,5 ---however many tiers you want. It could also be a radius around the moon and not the entire grav well.
Some random ability ideas:
Attack Drones: Attack drones are released from the moon to engage nearby enemy ships. (this is just a dumb idea i came up with to substitute hangars for those you perhaps would want it as a race specific abb for. Would work just like the cap ships "spawn bomber/figther" abbillities, just in higher number and perhaps a little tougher. I dont think it fit hypercorp outstandingly well though. Hehe. But it can be twisted in a variety of ways and could just as easily work with regular figthers/bombers. I just picked the name attack drones because i felt they could have a limited power source to justify their limited duration.
Another alternative of this would be too actually spawn something bigger, like frigates or cruisers "Reserve Forces: Forces beeing held in reserve are deployed to the battlefield."
Hope something here can be used. Let me know if you want more elaboration/better text on a subject.
edit: i tend to forget the nano part of the hypercorp, i am always unsure how much you want to gear them towards nano tech and what that really implies, i am not 100% sure what nanotechnology implies, i know some see it as minature technology, but i think scientifically speaking that its far more advanced than that. In any case:::
Nanorobotics: Level 1: The Moon is host to a veritable swarm of tiny nano machines that are constantly dooing what they are programmed to do. In this particular case they repair any friendly ship in close proximity too the moon. Level 2: The Swarms of tiny nano machines take on a more offensive role, and attack any designated enemy vessel that comes within their range. (abbility can be active or passive, if active 1 tier work as some sort of repair, either single target or aoe, if passive it heals every vessel within a certain radius of the moon slowly : Tier 2 if active work as a single target or aoe damage and if pasive does damage to all enemy ships within a radius)
You could easily build tier 3 + aswell. for instance: tier 3: The tiny machines are now able to disrupt targeting systems by emitting tiny disruptive signals that are multipled by their numbers, in effect reducing enemies effective range and chanse to hit.
The ship building uppgrade for hypercorp could be called NanoForging or Nano Forge or whatever scientific term that applies.
Bah its so much harder coming up with suggestions than if i had done it myself, then i would just have gone my own way and thrown down the book hehe. Anyway Let me know what kind of stuff you want for the various factions so i can make more productive suggestions, right now i am stumbling a little in the dark.
Here are some words: Nano Repair, Nano Disruption, Nano Swarm, Nano Forge, Nano Medicine, Nano Weaponry.
