Welcome to the forts, good sir or madam.
As far as your wood problem... Now would be a good time to learn how to ration wood. Don't even THINK about going outside until you have some way of getting guys inside safely, AND a really good militia with high-quality weapons and armor, made out of ideally steel. You'll want to dig as deep as you can to get some magma, then forge some chains and a minecart out of magma-safe metal. There's a trick where if you install rollers in a corridor, put ramps on either side, engrave track, then flood it with magma and push a minecart through, it'll be filled with magma. You can then have the dwarves haul the minecart up to another floor, build a track stop, set it to dump on arrival, and that's a good way to get controlled amounts of magma from one place to another without risk of flooding. It's slow, but enough to get a magma forge going on higher z-levels. And as far as the caves, if you crack the caves open, then immediately build a wall so that nothing can get in, you'll still start growing trees where possible. So you should do that. (You'll likely hit the caves when searching magma anyways, unless you can dig right on the embark square lines, since I hear those will never hit the caves. Supposedly. I haven't tested it.)
It sounds like you got unlucky. A lot of, but not all, evil regions have "regional interactions", or an interaction tied to the biome that is applied to any legal target on regular intervals. The only ones currently in the game will turn things into zombies. The only way to keep zombies down for good is to do one of the following:
Bury the corpse in a coffin (I don't know if this is always possible, and it's resource draining--AND vulnerable to necromancers, though if you keep your tombs under tight lockdown it shouldn't be a problem.)
Burn the corpse in magma (Only works if the body is a body, and not an animated zombie--otherwise you'll just have a flaming zombie that won't ever get put out. You can install upright spears/spikes in a tile and submerge it in magma, then link it to an external lever, if you're careful, though.)
Crush the corpse under a drawbridge (works on animated zombies too, but is cheap)
Take the body to a tile that isn't classified as part of an "evil" biome (may or may not be possible, depending on where you embarked. I believe that caves on evil areas aren't always "evil" areas, but I'm not sure.)
Throw the zombie in a pit and seal it up (Will hurt your FPS since the creature is still alive, but it won't be able to do anything to you if you build the jail right. IE, with constructed walls/floors and drawbridges to control access. For that matter, cage traps work too, but have the same problem.)
Butcher the corpse (Does not work on sentients, obviously, and for it to be successful you also need to tan the skin and spin any hair/wool at a farmers workshop. Dropping a creature down a death chute and exploding them into limbs will NOT work, as individual limbs can get back up. Also, you should surround your butchery-related workshops with cage traps, just in case a corpse animates while the job is underway. In fact, put cage traps all over the place in choke points.)
I personally recommend bridges, since they're the hardest to screw up and the safest to use. You can build two raising bridges, one very close to your entrance, and one closer to the surface, after a long, winding tunnel. Link them to separate levers, have the far one pulling on repeat. Zombies should get smashed to paste. Anything that gets past, you should have time to pull the second lever, just once, to raise the bridge and block pathing, convincing the zombies to eventually leave the hall. Multiple crushing bridges along the inside of the hall will help clean out any stragglers that decide to stay in the tunnel, since they generally need a path to dwarves before they decide to do anything.
NINJA-EDIT: Yeah, that's right. I forgot. Unless you're burying your dead (which, as mentioned, is risky, not to mention the creatures can rise from the dead while you're carrying them to a coffin), you'll want to build stone slabs at a mason's shop, then have them engraved with the names of the fallen and then installed as furniture. This stops dwarves from coming back as ghosts, which have behaviors ranging from giving minor negative thoughts to those in their way, to organizing the occasional party (if you're lucky), or throwing dwarves around, chopping off limbs, and literally scaring them to death (if you're unlucky). Beware, as you CANNOT engrave slabs for caravan guards due to a bug. They wont' even show up in dead/deceased lists.