First Paragraph of the GS article seems to stumble over itself a little bit, perhaps try cleaning it up a little, like the below?
Gemini Sovereignty was a government founded sometime around 22,000 B.C.E. in the Earth timeline, united by their first Lairae Mihmaetes. Their primary strengths lie in medicine and the sciences, while their military prowess is significantly weaker, the Gemini being a primarily peaceful civilization. Within the Sovereignty, art, music, and history are valued very highly, and in some cases are given even greater status than the sciences that the Gemini are so renowned for.
The article starts by stating that the Geminites aren't very strong militarily, but then the history section goes into great detail about all of the national war that went on before they unified under a central government - surely such a lengthy period of fighting would mean that they would develop more militarily than scientifically, and it doesn't give any indications anywhere of any gradual change from one to another after they made peace with themselves. This also contradicts with the description of the military later on down. Also, as a minor note; you said that their First Contact was with the Romulans and it was recent, but yet they have various things in place for not starting wars with other powers, and for intelligence operations against enemies... surely if they're a unified government with no prior knowledge of alien life, then these things wouldn't be as prominent?
There's also a section in there called 'Geminite and Romulan War', but it's just a heading, with nothing in it.
The bit about the High Court is very confusing. It switches between saying that the Lairae selects the High Court, to it being selected by its own members, and some of its selection rules contradict and cross over itself.
In the bit about the Lower Courts, it says that empirical proof must be had before they will issue a verdict either way - which would mean that if the prosecution and defence cannot make an unquestionable case, then the case would go on for all eternity.
Other than those few points (and for some I think, part of them), I think that the article as a whole could probably do with a full going over to improve the flow of the sentences and paragraphs to make it read better - doing that would probably also make it sound less contradictory in the parts I've mentioned above for the most part. Other than that though, it has a lot of detail in there and is a really good start to work from.
For the Dalcium article:
The first sentence contradicts itself; "Dalcium is a mineral found on Gemini Beta in abundance and a rare mineral found on Gemini Beta." I'm not sure which one you meant to put 'Alpha' for, but I assume that it was just a simple mistake
I'm not sure about the bit about the drilling and it being in liquid form here? Could you describe that a bit better maybe?
Also, perhaps we could change the 'Vulcanoid' reference to 'life forms with copper-based blood'? Just makes it a bit more generic, considering their limited knowledge of galactic civilisations - and it's always plausible that on one of their worlds there were animals which had green blood like Romulans and Vulcans.
Other than that, and a bit of minor tweaking of the sentences to make it flow a bit better, it looks good
It could be worth explaining though why it only affects those with copper-based blood, and not iron-based? The current description doesn't seem to have any reason in it why it would only affect one type of life form but not others.