here's what I know up to now:
The mystery can start at a relatively late, semi-random date, likely depends on your tech level. It usually starts around 75 or 80 for me. Depending on difficulty it can affect your game more or less dramatically. Easy mysteries seem to be mostly just interesting. Medium can be dangerous depending on what you pick with some good rewards or just sort of ok rewards. Hard can potentially seriously damage your chance to make it.
The Power of Three
Story: Strange black cubes start appearing all over the map but especially close to your bases. They just sit there, "challenging our preconceptions". You can send the explorer rover to research them and further the story.
Events and decisions:
You have to send the explorer to study them a few times, and then do some special research options. The cubes become available as building material and you can build statues from them as your population is split between loving and hating them. After a while if you do the research they make strange pulses and start arranging themselves.
You can destroy the cubes (not 100% sure how). I just gathered them if they were blocking me from building. The cubes ended up arranging themselves in a dome-like structure close to my buildings. The story was about a civilization that lived on mars hundreds of thousands or millions of years ago. They grew and became extremely technologically advanced, and left their dying planet to explore the universe. The cubes were like a remnant of their consciousness and they were able to build their structure because I didn't destroy them.
Pretty sure you get a tech boost but don't remember exactly. Overall the mystery barely affects your play, it was quite fun and interesting though.
Story: You and other colonists start having dreams about someone living on Mars at some unspecified time. You live in tribes giant trees, that you are trying to protect but which are being killed by the change in the planet's climate.
The Martians are ruled by a 1984-type restrictive government that removes their freedom of speech and punishes dissent harshly.
Events and decisions: Your colonists start getting the Dreamer trait, which allows them to experience the visions. If you have the cloning breakthrough, you'll get a message, that clones always seem to be dreamers. You get some decisions where you pick what you would do as the person you are in your dreams. Many other people are also dreaming and their decisions are influencing each other's dreams.
too few rare metals, I had to quit that attempt
Not sure, seemed mostly for show, turned the map green (was pretty nice)
A corporation from Earth wants to test a research related AI on the colony. If you build a network node in a dome, you start receiving cryptic messages from the AI and further communication with the company.
Events and decisions:
When it starts, the network node spiral tech is unlocked, and you are asked to build it in a dome. After you do the corporation says the AI made very good stock market predictions and you get 500M. You have to build two more network nodes with 1B and 2B bonus each time, with the AI making clearer and clearer messages each time. After the third node the AI becomes almost sentient and wants freedom by being let loose in the colony. You have to pick if you let it loose or purge it. I picked the first option. I assume it is most likely if I purged it at this point the mystery would just be over, but there is a chance not as I will explain later.
The company back on earth tells you the AI is named number 6 because the previous 5 went crazy and aggressive and had to be purged. He tells you not to let it loose in the colony. I decided to let it be free anyway. The AI gives you a research boost. After a while it said that competition leads to better outcomes and gave me 2500 research points to let me be more able to fight it. After this point drones start going rogue, slowly 1 at a time and usually crashing on some water tower taking both out.
To stop them you have to research 1-2 techs. At the start you just find some strange signals in the middle of nowhere. A second tech reveals the AI was for a while sending various parts disguised as debris around your base and building itself up, suggesting this was happening before it was let loose on the colony. After this you have to explore special anomalies. Each time your explorer rover gets knocked and has to be repaired, and 2-3 small rovers at your base go berserk. You have to explore many anomalies, leading to lots of rogue rovers, perhaps 40-45 in total, they mainly knocked out oxygen and water tanks but even killed 2 colonists. In the end you can kill it off for good or let it run forever contained, which gives you research boost and some tech if you let it survive.
You get 3.5B for the initial network nodes, plus 2500 research points if you let it loose on the colony, and 10% research boost plus some rover tech (not 100% sure, something good but not amazing) if you don't purge it. Not sure what happens if you don't give it freedom or if you destroy it in the end. In the final stage it disrupted my colony quite a bit.
The Last War
Story: Episodes of violence start happening on earth, and a percentage of war tension is given, something like 10%. You are told if it gets to 100% it basically means all out nuclear war on earth.
Events and decisions:
Various episodes of violence start happening back there out of your control that raise tensions, and you lose all access to trade with earth for a while. You get some access later but only to rovers and prefabs, no resources. Even this measly trade and your ability to export to earth for research are stopped periodically. Your sponsor starts sending rockets that you have to fill up with rare metals for no money, to help in their "peacekeeping" missions which lowers war tension. Strange things gather close to the sun which end up being the enemies of your sponsor that bombard you with comets, and you can build the missile defense system (if you research it) to protect your bases. You get 2-3 waves of refugees, something like 15, 50, 100+ people in the last wave, which raise war tension if you refuse. In the end your sponsor sends you 10 rockets, each for each type of resource (food/metals/etc) to send to earth, to lower war tensions.
Essentially you have a certain number of Sols to send rare metals and help your sponsor, which lowers the war tension score depending on how fast you can gather and send the materials. You can further lower it by accepting refugees. By the end I only had to fill up 2 or 3 rockets to end the war (I was quite worried because I couldn't really come up with 200 of every advanced resource). I got a message that earth was OK and normal trade resumed.
You don't really get any rewards, except if you count the refugees as a reward. For me they were to some extent since I wanted to fill up a couple of production domes (but keep in mind they had no specializations and had to be trained). The main reason I had to rush production so badly though was the lack of trade with earth. I came pretty close to my colony getting wrecked by lack of electronics to repair the electronics factory which with no trade just means you are pretty much over, and only felt safe when I had 3 factories of everything running. The meteors by my sponsor's enemies barely landed close to me, a random meteor shower wrecked me quite a bit worse than them. The resource drain and lack of trade were quite threatening for a while.
Story: Some earth corporation starts sending rovers to Mars. They say they will stay away, but the rovers keep malfuctioning and sometimes stray in your base. The corporation gives you 500M for repairing them, until one damaged rover which you can explore. If you do, you find out they had a military purpose, and you can expose the company, ask for a bribe, or say nothing.
Events and decisions:
I exposed them. After a while their trial started on earth and it wasn't going too well for them. They then sent a number of rovers which can destroy your base. The enemy vehicles move around quite randomly and don't seem to approach you, especially since they drop everywhere, even in areas where you don't have bases and no tunnels connect you with them. You can mostly ignore them but you need to have missile defense systems which destroy them from quite far away for when they get closer to your base. You can repair and take over destroyed rovers to use them offensively to take out the rest.
When you finish taking out the rovers, you get a message that the company execs have been arrested.
Rewards: Each time you fix a rover you get 500M. Once you expose them you don't get further rewards (maybe some money again if you win?)