1. How did the Korath have such devastatingly high fire-power so quickly. They had fleets of 84 laser power totals for example, but total hit points of something stupid like 12! What was going on there?! I could only muster about 6-8 attack on medium hulls by then, so a fleet of about 24 power, but much bigger hit point total and still got slaughtered by them.
A few things can be responsible for this: The Korath (if I am not mistaken) start out with a bonus for weapons, which makes weapons they put on ships a bit more devastating. Large fleets of small ships can also really rack up the damagecounter if they have decent weaponry, key to this is a high logistics value. The AI probably rushed weapon tech and got very far ahead of you. If it aquired a much larger planetbase than you, it also most likely had a much more impressive researchbase, explaining tech leads on you.
2. The Korath spored Mars and turned it radio-active. When, why, or how did Saturn suddenly become a class 16 radio-active planet when it started as a class 0?!
This sounds like a bug, or an event. I know of an event that will change uninhabited planets into habitable ones, but if I recall correctly (I may not) it doesn't turn them into extreme planets. So my bet is something went wonky.
3, A minor race called the Snathians (I think) popped up in a video sequence which said something like they had stolen ALL the Korath techs (I had 'Advanced' infiltration on the Korath and Drath btw) and they would be glad to give them to us, but at a cost. They wanted 'Star Systems'. When you clicked 'done' after the video, nothing seemed to happen. What was all that about?! That video sequence came up at least twice, and each time I was none the wiser as to what was going on. What is/are 'Star Systems'?
Ah, sweet bad luck to you. This is a semi-broken event that has been inherited from galciv, but not properly implemented. An example of a star system is the Sol system (containing Earth and Mars). The original idea was that the Minor Race AI would severely increase the value it assinged to planets you offered it in the Diplomacy screen, therefore making it possible for you to trade Star Systems for the techs it acquired from the Korath (provided tech trading has not been turned off). However, although the minor race (in this case the Snathi) DOES receive the techs that the Korath have researched, you will have a very hard time making them accept planets for it in return as they pretty much regard them as worthless. (Which would be a stupid trade on your par anyhow, seeing how low you were on planets to begin with.) Trading the techs for other techs or money is still viable though, except in some cases weapons techs: if any AI considers your military might to be too formidable, it will refuse to trade weapons tech to you.
So, apart from the three questions above, generally, what does one do (apart from Ctrl-N) when penned into a corner with no extra colonies?
Was it not a good idea to colonise the other two planets that I did?
Starting in such a poor starting location can be reason to give up immediatly, but it can also give rise to the more awesome underdog games. What you have to do ("somehow") is come out on top in the end. Just how to do this depends VERY much on the situation, but here are some pointers:
I'm surprised you even managed to grab those two extra planets, but I think you were probably wise to do so as they significantly increased your research and production capability. However, they are also a liability. Grabbing a planet in Drath and Korath territory is something that most likely increased the chances of the wars you were dragged into. The AI does not like conflicting culture boundaries, but this is not a problem that cannot be overcome.
Butter up your neighbours! Whenever the Drath and/or the Korath issue a threat and you know you cannot easily beat them: give in and buy yourself time.
Make them too busy to start a fight: Pay the biggest warrace in the galaxy to go to war with everyone else (in seperate deals), before they start wars on their own. Continue to instigate wars as long and frequently as possible.
Trade is a weak economic boost on the larger galaxies, but on medium it is actually pretty decent, not in the least because it provides a relations bonus. However, as you noticed the relations bonus is no guarantee AND trade is very vulnerable. You simply cannot rely on it unless you are certain you can stay out of all wars, or really have no other choice. It is a supplementary income at best, because it is so fragile. To me it does look like a risk you needed to take, though.
Diplomacy for the win! Trade for planets. Races that are on the losing side of a war are desperate for warships, there comes a point where they will sell you planets in return for your ships. Finding this point is difficult and requires you to have a decent fleet of your own, which appeared to be a problem last game. There is also a (rather lame) way to haggle races out of new planets very early in the game, I consider this method "cheese" but others disagree (as they have every right to do). A high diplomacy gets you better deals ánd provides a relations bonus.
Be oppertunistic. See a weak target? Kick it while it's down! Either directly join in the war against the strongest opponent, providing a united front and dividing the enemy strength, stay on the sidelines sponsoring the people at war with the strongest opponent while you try to outtech it (very difficult in your situation), or join with the strongest opponent and do better than him! One of the AI weaknesses is developing a comprehensive invasion strategy. This is a field where the human player can easely outperform the AI. (This resembles what you did to the Alterians, but next time be more thorough.) Get the Korath to go to war with someone, make sure you stay at peace with the Korath, send your transports in behind the Korath offensive fleets and snatch up the planets that they have cleared of hostile ships, before the Korath get their own filthy paws on them. Granted, it is ironically the Korath that are the hardest race to do this with, as they are pretty enthousiastic about their sporeships, much more than regular races are about transports.
You can also turtle up and go for a technology victory, but this may be difficult, depending on just how far behind in planetcount you were.
But whatever you do, you must remember that a poor starting location can mean the game feels a couple of difficulties higher than it really is. Therefore, if you are playing above the level you are used to AND end up in such a poor location, you are in for a really difficult game. It is not surprising you lost, considering your galciv experience so far. I'd have been surprised if you had pulled it off first time around actually. It is the lost games that usually teach you the most though and they can still be pretty entertaining!
Good luck next time. 
