How long can I store cooking wine

Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)

See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."

So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.

• Keep your wines between 45-65 oF (7-18 oC). Wine that is too hot will age faster and frozen wine can expand and push out corks or shatter bottles.
• Keep the temperature relatively consistent: avoid large swings in temperature.
• Keep the wine out of UV light (sunlight and some types of fluorescent light) which will age the wines faster.
• Humidity levels shouldn't be so low that the corks dry out or so high that mold can form. Keep between 50-80% roughly.
• If the wines are stoppered with cork, keep the bottles on their side to keep the corks moist. You don't want the corks to dry out and lose their seal. Bottles stoppered with screw caps or plastic cork don't need this treatment. However storing bottles on their side is pretty space efficient. (Also, it really sucks to remove a dry cork from a bottle of wine, which can be a problem if you buy cheap wine from places like Trader Joes where they are not always stored properly.)
• Keep the wines away from vibration. (This is why storing wine on top of a fridge is a horrible idea, it's hot and shaky up there.) Excess vibrations can disturb sediment.
• I've also read that you should keep wines away from strong odors because corks are porous. So maybe don't store garlic and vinegar with your wine collection.
• Enclosure shouldn't matter a huge deal... if your wine stoppered with a cork is tainted, it was tainted before you aged it in your cellar for such a long time. Screw caps and synthetic corks prevent the taint from happening in the first place.
• You don't need to rotate your bottles, which disturbs sediment.
• Not all wines age well, especially many whites. (Although there are definitely whites that do age well.) I have suffered the problem of storing many of my own Sav Blancs for several years without tasting to make sure they were still good, then having to pour out several bottles when they were over aged. So if you have many bottles of a single wine, drink a bottle every now and again to see how it's holding up.
• Finally, here's an interesting article I just read about aging champagne. Apparently it's not always a bad idea.