How can I start up an effective metal industry?

Alrighty, in order:

Why don't I see more stones deposited in my stockpiles?

Most of the "stones" you mention having mined are, in fact, raw gems. The others should show up in your stone stockpile, if you have enough room. (If it's not on this list, it's a stone).

Which of the stones that I've struck so far are useful for crafting picks, weapons, and armor?

Which stones should I prioritize finding to craft the aforementioned picks, weapons, and armor?

Tetrahedrite is the only one, as it is an ore of Copper, which is (the weakest) weapons-grade material.

What should I do with the other stones that I find?

"Common Stone" is all of equivalent value. You may be able to make some interestingly colored furniture out of, say, your Kimberlite, but color is the only difference, which is moot from a gameplay standpoint.

So, do with it whatever you do with the rest of your non-announced stone!

The gems, however, can be used in your Gem Industry.

What's the easiest way to transport, store, and keep track of all my stones and metals?

Unfortunately, there really isn't. Every stone must be moved manually, and if you've a lot of stone ( and few idle dwarves), this might take a very long time. Consider a Quantum Stockpile if you're desperate.

Only those two stone types are ores. The others are simply stone. Most of what you have found is used only for building or for making into cut gems. Tetrahedrite will produce Copper: the most basic of useful metals. Gold is heavy and soft, which makes it ill suited for armor and weapons. It is mostly used for decoration. The only stone that can be used to make weapons is Obsidian, which can be turned into shortswords.

You should look at this page to find out which stones you can turn into usable metals. Looking at this page will show you which metals are best suited for which purposes.

Other stones you find should be stored near your Masons/Craftsdwarves/Mechanics, as they use stone in their work. For the most part you can just leave most of the stone lying around your mine, as it doesn't get in the way. Dwarves will transport stone as needed to keep your stockpiles full. If you want to see what types of stone you have, press z, go to the Stocks header, and scroll down to Stones. Unless you are going for some specific color coding (which is usually a bad idea when just starting out), you shouldn't really worry about what sort of stone your dwarves use.

Looking under the Stone heading on the z-screen will show you which stones are considered economic. This will include all ores, as well as a few other stone types. By default, your dwarves will not use ores for crafting, and will instead save them to be smelted, so that's not something you have to worry about. Opening up the Add Task menu on your smelter will show you which ores you have available, and what type of metal that reaction will create.