In space, use the merchants to get fuel and evade combat. Watch out for the space station and dreadnaught as they both instagib. Slip in to the planet, and try to have 50+ armor left when you do in case the cannons on the planet hit you.
The math thing is funny, and obviously the third question cannot kill you no matter which you pick. Out of curiosity, I discovered that the first two questions can indeed kill you and give you a funny message telling you the answer and that you might want to write it down.
When landing, I found that going into the city was ironically easiest. Having a larger landing pad gives a bigger target and a wider area of guaranteed no AA weapons near your target, both of which are nice. The biggest thing is to remember to use the up arrow to control your descent. When you're first getting the hang of the mode, I recommend clearing out the guns near the landing zone and hovering while you land.
The car scene has its own challenges. Going the back roads is IMO the hardest, make sure you keep shooting and swerving to give yourself space to dodge the mines (which instakill you). Someone said the highway is impossible, but I didn't think it was that bad. The bombs are heavily telegraphed (airplane flies overhead, dots appear on the ground). Only the bombs on the far left can hit you, so don't even worry about the rest. Just make sure you move your car when the dots appear, in case one was hiding under you.
What seems to be the hardest game of all (the base raid) is actually one of the easiest once you figure it out. The sewers are the fastest (if you're a marine, haven't tried it otherwise). You have enough health to clear out all the enemies, so just keep marching and fighting. The front door is suicide as far as I can tell, there's no way to crouch or aim or anything so you're just hoping for impossible luck. Bluffing and taking the inner guards is easier than it seems when you figure it out. Crouching with crap gives you cover, you can tell it's working because there's a green arrow pointed at you. If you go too far past the edge of the crap then you lose cover, same goes for the enemies. Other than that, if the crap is fully between both you and your opponent neither of you can kill the other one. On paper, that means it's not going to alter the damage scales at all...
I tried taking cover and shooting, and eventually everyone but one person would die. While I initially thought that there was a chance of hitting a target while still under cover, I eventually realized what was actually happening, and it's flipping hilarious. The
dumb incredibly intelligent AI shoots at you constantly the moment you are in range, despite you being completely invincible behind cover. They do this even if there are more than one of them. Perhaps most importantly of all, Tarn did NOT disable friendly fire.
But it gets even better. The AI will
foolishly boldly march to the left to slay the intruder (you), stopping only when they are in range. They do this across the entire map, meaning you can sit behind the first piece of crap you find and wait for the entire defense force to come up and kill each other one at a time until only one remains. In fact, there are always exactly 9 guards, so you're safe to kill the last one once 8 of them have fallen to their comrade's bloodthirstiness. Just be wary, as occasionally a guard will eventually walk past the cover at which point they can and will hit you, so be ready to shoot them if they do.
For the final minigame, just keep your eyes on the prize. Picking the wrong one is a fatal mistake.