What is the best way to keep track of all the dwarfs skills?
Either V -> g on a Dwarf, or use something like Dwarf Therapist (see modding section of the forum)
If I rename a profession will they be renamed again automatically if they gain enough skill in a certain skill?
No. If you rename a dwarf's profession to 'Builder', they will stay "Builder" until you rename them again.
What does everyone else do to make sure that you have enough of each type of skill?
Depends on what I'm going to do and how much of it I'm going to do. That falls to playstyle.
As for your layout, it could work, but I'd lower the amount and specialize a few dwarves. For instance, you want more Furnace Operators than you want Weaponsmith. Unlike Smelting, Weaponsmithing, Armorsmithing, and MEtalsmithing, let's say, have a dramatic effect on a dwarf becoming Legendary in the skill. (Furnace Operating has a strong effect, but it's speed only, bars don't have quality levels. Weapons and armors do.) So you should consider "Training" in your plans.
Sure, you could have 5 competent smelters/weaponsmith/armorsmith, but wouldn't it be better to have 3 smelters, 1 legendary weaponsmith and 1 legendary armorsmith?
Unless you're hunting and butchering many, many animals all at once, one butcher and one tanner should do. You do want several masons, though, if you'll build towers/megaprojects. But if you won't build those, then you don't really need all that many, now do you?
Also consider your hauling needs. The more you produce, the more haulers you need to keep up with the workload.
Also, how does everyone name their dwarfs? I have just been numbering them to keep track of who I went through and assigned skills to. Is there a better way?
Personally, I don't give then nicknames. I rename everyone's professions into "Groups" (like Foodworkers, Leatherworkers, Clothiers, Metalworkers, Builders, etc) and use that. Though you could nickname them based on their *primary* skill. Or don't nickname them, it's all based on what works for you.
Cheers.