Ok, I'll address the easy ones first:
Should I be mining near water?
For small bodies of water it makes no real difference one way or the other. That said, you have fewer mineable levels available if you're beneath an Ocean or Deep Ocean, so you may cut into some of the ore distribution ranges if you mine near Oceans. This will generally only affect Coal and Iron though.
Should I find a village and mine near it?
Again, the presence of a village makes no real difference to the underground ore distribution.
Does it matter if I mine during the day or night?
Yes and no. The time of day has zero impact on ore distribution. However, you will find it easier to mine during the night because you will encounter fewer enemies. During the day (unless it's raining) enemies will only spawn underground, whereas at night they will spawn underground and on the surface. Because there is a limit to the number of mobs that can be spawned, you'll run into fewer enemies underground at night time due to the greatly increased area in which they can spawn.
Should I use TNT near deposits or strictly stick to the pick axe?
Do not use TNT to mine. It's expensive and you'll destroy a considerable portion of your yield each time.
How far down can I start to expect to find better items?
I'm assuming a definition here; I'm assuming that "items" means "ore".
With that said, here is a nice graph of ore distributions in vanilla Minecraft (credit goes to Reddit user cryzed- for this graph):
Disclaimer: this graph is based on version 1.7.10, so may not be accurate for 1.8+.
It's a little hard to see on the shrunk down version of the graph, but I'll go through the gist of it and you can click through to a much larger version if you like.
A quick note on terminology here; "level" means "blocks above bedrock". So "level 5" means "5 blocks above bedrock".
Coal Ore: You'll run into tons of this at most levels. It's most common between 5 and 46 levels above bedrock (and consistently so), but you can find it at almost any height. It gets much less common above level 58.
Iron Ore: As with coal, you'll find this at most levels, it's most common between 5 and 51 levels above bedrock, then drops off rapidly until level 63. You can't find it above level 63.
Redstone Ore: Redstone has a much narrower band than Coal or Iron, spawning between levels 1 and 15 only. It's most common between levels 5 and 13.
Gold Ore: Gold is most common between levels 5 and 29, and cannot be found above level 31. It's common enough that you should still find a fair amount of it (but Redstone is ~6.5 times more abundant).
Lapis Lazuli Ore: Lapis is most common between levels 10 and 21, tapering off gradually until level 30, above which it cannot be found. It's slightly less common than gold.
Diamond Ore: Diamond peaks between levels 5 and 13. It's slightly more abundant between 5 and 8 than 11 to 13, however you'll find frequent lava lakes around level 10, so the 11-13 band is much safer to mine in.
Emerald Ore: Emerald is a funny one. It can only be found in the various types of Extreme Hills biomes, and only spawns in single block veins. If you're based in an Extreme Hill-type biome, you'll find a reasonable amount of it, and it has roughly the same spawn range as gold (5-31), but it's still not common. The easiest way I've found to mine Emeralds is to find natural cave systems in Extreme Hills and just explore the sub-30 levels looking for exposed blocks.
Another way to increase your mining yields is to use a pickaxe enchanted with either Fortune I, Fortune II or Fortune III. These enchants give a 33/25/20% chance to drop an extra 2/3/4 items respectively (yes, Fortune III procs less than Fortune I, but still works out better than I or II).
For mining strategies, refer to What's the most efficient mining strategy? for some good tips.