Minecraft – Iron Golem Farm not working

Most of what you're asking can be found on a couple of articles on the minecraft wiki site, but I'll summarize the big points:

• Villages are specified by the existence of doors that meet certain conditions. For a door to be counted as "in a village," the door must have more blocks on the outside than on the inside and a villager must exist nearby - no more than 16 blocks horizontally and 3 to 4 blocks vertically. A block is counted as inside if it is shaded from above. A block is counted as outside if the sun hits it from above. Here are some common techniques for registering doors.

• If the distance requirement for a villager nearby a door is not met, the door can become unregistered after some time. That can be resolved by destroying it and replacing it.

• Each door can support .35 of a villager, i.e. you need about 3 valid doors for each villager (more precisely, every 20 doors support 7 villagers).

• The boundary condition seems a bit hazy, but the 16 block requirement in the first point seems to imply that it's 16 blocks from the most outlying door. However, villagers can move up to 40 blocks beyond that before they actually unregister from their village.

• You must have at least two villagers if you want to start an entirely new village. That means you've either got to find a village and transplant villagers from there, or cure Zombie Villagers using a Splash potion of Weakness, Golden Apples and time.

  • Once you have two villagers, you can have them create new villagers by "making love" (villagers are genderless, so any 2 will do).

  • Villagers will attempt to birth new ones when the village can support more than the current number, as per the third point (doors can support .35 of a villager).

• Iron golems spawn when there are at least 16 adult villagers and at least 21 valid doors.

• Iron golems and villagers both will wander aimlessly if they are not created naturally or are not in a village, respectively.

This information is valid as of snapshot 12w21a, version 1.3.1.

EDIT: To answer the more specific questions OP had...

I don't think anyone really knows, except for the developers, where the hard cutoff for the village is, or if this is one. Apparently at some point, Notch was going to put in some physical boundaries for the villages, but they might not even exist in newer releases.

This mod claims you average out all the locations of the valid doors for the village center, then that the boundary is the largest distance from the center to the farthest house (or 32, if there are no other houses) but it's not clear whether the mod maker thought of these arbitrary boundaries or whether they're actually in the game. I'm going to stick with what the wiki implies and say that villagers register to a village and probably wander up to 16 blocks beyond the outermost doors, and only deregister from the village if they're beyond 40 blocks.

Not completely, no. You will need some sort of air shaft.

Villages, in Minecraft, are just a collection of doors that meet the condition of having more spaces on one side exposed to sunlight (This page on the Minecraft Wiki goes into the details). This means that a village/Iron Golem farm without any skylights won't work, as none of the doors placed will count as Village houses. The closest you can come will involve either open shafts, or else glass or something similar in the ceiling.