How on earth (or whatever planet it is) do I win, or even survive, on level 3?

This one gave me a ton of trouble as well. Happily, I found a simple way to win. It takes a little bit of prep time, but not long at all, compared to most missions.

First off, go ahead and take out the first biter nest. You don't need anything special for it, just run in holding space bar, and take your licks. After you get your explosives and drones, retreat back to the original location of the car. Once you're there, Biters won't come and attack you, allowing you to do what you need to in peace.

Now, retrieve your supplies from your car (yeah, I forgot it had stuff, too). Your goal is to make something like 20 more defender drones. Not hard, just time consuming. You can make it easy by right clicking on the defender drones in your build menu. That will add five items per click. (Incidentally, that works for any item). Once complete, put them in an easily accessible shortcut on your quickbar.

Now, save. Take the car down the path by the lakes; there should be a thin path between them. Right at the bottom, turn east, and give her as best you can. The Biters and Worms will make short work of your car, but that's fine; it's only meant as extra health. Once it explodes, start dropping your drones, and retreat into the trees. Do NOT engage the Biters yourself; it slows you down too much. The idea is to get into the trees, and let the drones take care of them. The ship is just in the southeast corner of the biter nests; all you need to do is get there, and grab the computer. That will end the mission.

Frank's answer is the sneaky way. The brute-force way (and probably the one that the devs had in mind) is: wall off and fortify a choke point, then build a factory complex to churn out rockets and defender capsules. (And I suppose grenades if you like those; I wasn't impressed by them.) Everything you need is in your starting inventory, the trunk of your car, or the crashed ship you salvage at the beginning of the level. If you're careful, you won't need to build anything in hand.

I ran out of explosives and iron about the same time my turrets ran out of ammo. At that point, I had something like 70 rockets and 200 defender capsules, which is enough to pretty much just go Leeroy Jenkins on the biter hive. Keep 30-40 defenders out at all times, and keep moving, and not much will be able to hurt you; bombard the worms from long range with rockets, then move in and wipe everything out with your SMG and defenders.

I beat this one in 39 minutes. It was tedious, but what I did was slowly advanced forward with turrets defended by walls. Once I advanced into a new area, I would deconstruct the previous one and use that to move forward. Only thing you gotta build are repair kits. Wasn't sure if I'd make it through with enough bullets, was down to about 250 when the last nest was killed.