The build begins (finally)...
Of course it started raining, and I flipped the positions of the granite and diorite blocks (purely cosmetic, I know, but I wanted the granite blocks as the spawn blocks). Fortunately I haven't built much yet, so I guess I'll start over.
Okay, things are going well after a bit of a rough start. One layer finished, four more to go.
And look, I caught a golem falling from one of the upper village spawners! 
Once the first layer is in, it's a piece of cake. Two more layers in less time than the first one took.
Each 16x16 layer takes exactly what you see in my hotbar below.
2 stacks each of building blocks and string, plus 1 stack each of cobble walls, slabs, sticky pistons, tripwire hooks and trapdoors. Technically you could eliminate one row of (16) blocks above me in the picture, but I wanted the farm to be 16x16 (not 16x15) for the creeper face facade.
Alright, the creeper spawning mechanism is done. Time for an aerial inspection to make sure I didn't leave any temporary blocks behind.
Looks good. Just need to build the enclosure and it's done. But I'm hungry, so that's all for now.
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"You keep using that word. I do not think it means what you think it means."
-- Inigo Montoya, The Princess Bride
What, exactly, do you guys all mean when you say "strip mine"? Because I don't think you're using it right. When I hear "strip mine," I think of this:
That is an actual, real-life "strip mine." The term refers, not to mining in "strips" (that's your plain-old, everyday "branch" mine -- you dig one main tunnel, or "strip" if you insist [but I wouldn't, because that road only leads to just this sort of confusion], with more tunnels occasionally "branching off" to the sides at regular intervals), but rather, it refers to "stripping" everything away, leaving nothing behind except for a giant hole in the ground. In Minecraft, this is commonly known as a "quarry" (another type of open-pit mine, in real life), and it is a horribly inefficient method of acquiring resources other than cobblestone.
The best method to find resources is the one that exposes the largest number of blocks in the least amount of time (or, arguably, with the least amount of wear on your tools. But since using the tools takes time as well, it pretty much boils down to the same thing.) Ore blocks, though randomly distributed, make up a certain set percentage of all stone at the appropriate depths. So the more blocks you are able to look at, the more resources you will find among them. Usually, this means caving is the way to go, if all you're looking for is coal and iron, or even gold if you find a deep one. There are hundreds upon hundreds of already-exposed blocks in the cave walls, so you don't even have to dig anything but the goodies!
But it's hard to find caves that go deep enough to reach diamond level, and when you do, they're usually partially or completely filled with lava. The best way to find diamonds, then, (or redstone, which appears at the same depths as diamond, only much more frequently), is to branch mine with a 1x2 tunnel (three new blocks exposed for each one broken; four, if you count the one behind it, but that one you were going to break next, anyway, so it doesn't really count) at y=11 (feet level, eyes=12.6) so that the ground you are walking on is level with the tops of the lava lakes. There will be no pits for you to fall into or have to navigate across or around, because all airspace below this level is replaced with lava. When you come across a lava lake, you can just pour out your water bucket next to it, scoop it back up, and then walk safely across your nice, new obsidian floor. And you will have very little lava come pouring into your tunnel from above or to the side, because the majority of it is below you. All you have to worry about in that regard are a few stray source blocks here and there, and the occasional higher-level pool.
I haven't had to go mining in quite a while, but the last time I did, I used Phoenix's X-Mining technique. It's a lot like branch mining, only it cuts down on the time you have to spend backtracking through the tunnels you've already dug out.
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But the dirt blocks themselves are not unlimited. Brobdingnagian in number, perhaps, but not unlimited. Being a non-renewable resource, then given enough time and determination, you could theoretically run out of them. You could mine every dirt block in the map and then there would be no more dirt blocks. Seriously, people, I see this argument just about every other week -- you guys need to look up the definition of the word "infinite." It doesn't just mean "there's a whole lot of them." It means "literally unbounded in number or scale."
The Minecraft world is not literally infinite. It simply can not be, since that would take a computer with infinite processing power, which itself would require an infinite amount of space. And even the best scientists in the world don't even know if there is an infinite amount of space in the entire universe or not. But even if there is, it's certainly not completely filled with computer circuits running a game program.
I repeat, the Minecraft world is not infinite. It has an edge, even coded into the game there is a hard limit past which you can not travel, which occurs about 30 million blocks out from where you spawn near the center of the map.
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A mob can only spawn in a certain position if the block below the spawn block has a solid top surface. Blocks with a slab in the top half of the block-space have a solid top surface. Blocks which only have a slab in the bottom half, do not.
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Yes, as indicated by Herb in the post above mine, your problem is that you have the gamerule mobGriefing disabled. This means that endermen and creeper explosions won't wreck your builds; it also means that villagers won't plant or harvest crops, or pick up food items that you leave for them. This means that the only way they can become "willing" to breed is by a player trading with them (100% chance they will become willing the first time you perform a particular trade offer; 20% chance of success each time you perform that particular trade afterwards.) If you want them to breed automatically, you're gonna have to bite the bullet and enable mobGriefing.
Your world doesn't have commands enabled. You can enable them temporarily by pausing the game and select "Open to LAN". Select "Allow commands: Yes" and click "Ok." Type "/gamerule mobGriefing true" (watch the capital G in mobGriefing), then exit and restart the game. The commands will again be disabled, but any change(s) you made while they were in effect will remain, unless and until you change them back in the same manner.
If you want commands enabled permanently, you can download a tool called NBT Explorer, and use it to edit the "allowCommands" value in your world's level.dat file from 0 (false) to 1 (true).
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You transposed your numbers when calculating. Your overworld portal is at z=6756 but your nether portal points to 822*8=6576. This is 180 blocks away from the existing portal, and outside of search range, so a new one is created near the entry point. Move it an an additional 22/23 blocks to 844/845 like VolcanoBomber said and it should work just fine. It will point to 6752 or 6760, a mere four blocks away from the existing portal and well within search range.
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Stop eating poisonous foods. Cook your meat, feed the zombie flesh to your dogs, throw out the green potatoes, and only use puffer fish and spider eyes for making potions.
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A farming villager must either be inside a village, or else more than 32 blocks outside its boundary (for most villages, this means either less than 32 blocks from center, or else more than 64.) What you don't want, is to have him in the zone between these two regions; there, he'll be outside the village, but still near enough to know it's there, so he'll try to run back towards the village until he is inside the boundary. This is prioritized above all other actions in their AI routine, which is why I refer to this state as "panic mode" since the villager can do nothing else in this state and if they can't pathfind back to the village, they'll just run around in circles until you fix it.
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You're already doing it. Go deep, and dig, and dig, and keep digging till you find them. There is no magic "trick" to make them appear higher up or more frequently.
You can increase the ratio of blocks viewed:blocks mined by mining a narrow 2-high tunnel in a straight line for ever and ever which will give you the highest yield of diamonds discovered versus blocks broken, but the downside is that by the time you are done, you have traveled very far away from where you started. You can mitigate this somewhat by doubling back on yourself, but this does hinder your efficiency somewhat as every time you turn a corner, you are breaking into a section that has already been revealed, so you are gaining no new information (or less of it, anyway, there's still new blocks behind the ones you broke even if you already knew what the ones to the sides were.) Make sure to have more than just 1 or 2 blocks in between your tunnels, then, or else you will be wasting significant amounts of time "rediscovering" veins that you've already known about. Three blocks or more between tunnels is best. As you space them out even farther, you get even higher efficiency (in exchange for less fully-complete coverage) but with diminishing returns, up to about six or so blocks spacing when it hits a maximum, and increasing the spacing beyond that point has no benefit.
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It's actually supposed to pop when they step on the stone pressure plate next to the crop. You do have the pressure plate there, yeah? It ticks the blocks next to it and the center crop should pop when it checks the lighting. Without the pressure plate, the crop might not receive a random tick for several minutes.
If you have the pressure plate and it's still not working, whats the light level above that center block? (You'll have to replace the farmland temporarily to check, or else it'll just read zero since it's not a full-height block and you'll sink down in by 1/16.) It should be reading 0 sky and 7 block.
Have you finished the roof yet? During construction, or without an opaque roof, sky light entering through the central drop shaft area will keep them from popping (even at night.) Otherwise, double check and triple check your light-source placement. I have the same farm in my world and it's still working in the latest release version 1.11.2
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No, but to activate their boost effects and not just the visual beam you do need a resource item such as a gem or ingot, which is probably what they meant.
The beacon power will remain active until the beacon block is removed or you want to switch it to a different power, which will cost an additional resource item at that time. Otherwise, yeah, forever.