Random events - how do I even fight those damned space fiends..
If you're speaking of "Unknown menaces", I have quite a lot of experience with them, since I mostly play with 200% randoms. IMO all the random "enemies" are easy, when you learn how to deal with them, but most of them need a different approach. In the early game are the Humans and the Liir in the worst "shape" to deal with them. Because of their low combat speed they can't run away where that'd save them. BTW where I say Lasers, I mean at least Green Laaers. Red ones are a bit too weak for the task.
So let's deal with those menaces.
Run away to the left of the planet and click peaceful resolve in the lower left corner. To eliminate it, get back with a fleet of laser-equipped DEs and autoresolve (game will try to strip them off weapons first, before outright destroying). ADs with AM torpedos and other heavy guns need a turn of manual "stripping" (click on the "Fire only at my target" button), before autoresolve will yield large research bonus.
Run away to the left of the planet and click peaceful resolve in the lower left corner. If AM is level-2, you may not get away. Thereafter research special project to gain control of it. In my low-econ games it usually takes 20-30 turns to finish it.
A nasty one. You need two full DE fleets with Light Emitters to have a chance to clear it, or ~3 kamikazee Rafineries and a fleet of laser-equipped DEs to clear it. For some reason the point defence stops working on drones after about a minute of combat, and only lasers keep firing. In any case expect heavy losses, so this one is usually not worth clearing out. You'll know that the Alien Wreckage is "close" around, when you start getting
Those happen only if you "visit" the planet with a fleet without Deep Scan section, or a fleet of less than 12 command points. So a fleet of 6 DEs or 2 CRs is safe, and also "removes" those cloaked ships. With a fast ship you can also run away. Just turn right and start running. This also "removes" them.
Have 2 cheap CnC DEs in your mining fleet. When you get control of the ships, click "Retreat from battle" (upper left), unpause, then select all ships and click "Retreat to the last line" (upper right).
Early game colonize with only one DE, then follow with more. When you have CR Biomes, equip them with gauss and laser PD in small slots, and fight manually. But usally it is cheaper and faster to sacrifice one DE colonizer and autoresolve.
The scout is lost. To eliminate space bees, get back with a full fleet of laser-equipped or Light Emitter DEs and autoresolve. Expect moderate losses. If you'll fight manually, you'll have less losses (don't move ships until the drones are cleared), but you'll have problems with the Larva (if it appears), because it's covered with Improved Reflective Coating and has rear-mounted Medium Emitter.
More drones plus the Queen. Queen doesn't turn, and fires only from her head, so approach her from the side or rear. It also likes to hide in a shadow of an asteroid. Use the same recipie as for Swarm Infestation, just have more ships. Satellites will also help to deal with drones.
If you fight regularly, a rare event. IMO not worth organizing defenses just against them. A full ring of small+medium sattellites plus good warheads will destroy them, but most of the time you'll still lose pop.
Same as Slavers. Sensitive only to energy weapons, but also to energy-based point defences. Appear only in games with Humans and/or Zuul.
Von Neumann Mothership
Early game: have 2 cheapest possible sacrifical DEs in orbit of each planet. When Mothership comes, send them forth for drones to "eat", and they'll leave.
Mid game: 3 DEs with accurate weapons (PD doesn't work!) to destroy drones. If you destroy also the Mothership, you have a fair chance that planet will be visited by
Von Neumann Berserker
A nasty one. Comes 4 turns after you've destroyed the VN Mothership. You need full rings of small+medium sateliltes + full CR CnC fleet of DEs + Deep Scan ship(s) to deal with it.
OR (cheesy tactic) 8 DEs with Leap Mines and an DE CnC with Deep Scan. If you'll destroy also the Berserker, you'll get a moderate chance to be visited by something significantly nastier.