Do races move ?
Nope. They just sit on their homeworld forever.
Are there racial differences ?
Will planets be unique enough without a unique race ?
Does it matter? If there are 100 planets and like 10-20 at most inhabited (one planet per race) it's a rare thing for a planet to have them... Either way I doubt it would add a lot of uniqueness to planets in general since there are so few of such races.
The low quantity of these really bothers me... I simply can't make 40 different races (which would make 40% of planets neutral which would be the best in my opinion).
Neutral planets are a fitting addition. If I'm grasping the whole of your concept, they represent a "resource" that is not so much contested but unaligned
Well, yes... Sort of. One of their uses is a neutral "cannon fodder" since they are attacked by all sort of invading aliens, symbiots, pirates, and since these neutral can put some sort of fight they can slow down the enemy (which happen to be your enemy as well). Anything that try to attack you will attack neutrals as well (maybe with few exceptions).
This certainly can lead to competition to control that resource
Competition with whom? There are no other empires.
As I reflect on your game I can't help but to think of the Crusades, particularly the First through Third. I suggest you read some sort of brief history of the ebb and flow of that sad conflict for some ideas. The Islamic world shared a common faith, but they were often at odds with each other, allowing invading armies to conquer in the face of daunting odds. It seems to exemplify the idea of how neutral planets (in this case cities) relate to the surrounding power struggle.
Yeah, sort of. Europe had the Church with a single Pope and, regardless of numerous local disputes and everything it was a centrally governed religion (to an extend) with some common goals for all kingdoms. Muslims had no such thing, they were all separated.
Yes, it's kind of the same here, the player is the Empire, all planets under one rule. Neutrals are divided.
But, more importantly, in the log run the neutrals will have to join to empire if they want to survive (invasion of aliens from another dimension that want to annihilate everyone). Also, neutrals share the same enemies with the empire.