Use a vegetable peeler to peel of large strips of rind. That way, it'll be easier to remove once you're done steeping. If you don't mind it being mildly alcoholic, steep them in vodka and then strain the vodka into the lemonade.
You aren't actually trying to cook anything. When you heat a solution, it makes dissolving a solid in to that solution much easier. So you can dissolve more sugar in to hot water than cold water. With a 1:1 ratio, you wouldn't be able to get all the sugar in to solution with cold water. So, you heat the water to allow more sugar to become part of the solution. So the answer is, as soon as there is no visible sugar in the water, you are done.
If you were to cook it for longer, you would reduce the syrup, increasing the concentration of sugar in the solution. You'd have to reduce it an awful lot before you burnt this. However, if you reduce it too much, then the liquid will become solid when it is cooled. Both of those would probably qualify as "over cooked".
Here's a nice science description of sugar solubility. Interestingly, they say that sugar has such high solubility in water that you can get 1800g in to 1L. Using this volume to weight conversion site, that's approximately 7.5 cups of sugar in 1L of water. 1L = 4.2 cups, so you can get ~1.8 Cups of sugar in to 1 cup of water (under ideal circumstances). So, if you are actually interested in cooking your water off to get a more sugary syrup, you can't reduce the water to less than 1 / 1.8 = or 55% of the original volume. However, if you do want a ratio higher that 1:1, then I would suggest introducing the appropriate amount of sugar in the first place and stopping when the sugar is dissolved rather than using less sugar and cooking the liquid down, as the first approach is much more precise.
(for purposes of this answer, I have ignored the possibility of super saturating the syrup, as that would be essentially useless for cooking....although it would be funny)
Best Answer
Simple Syrup is sugar that is has been dissolved in water. Heating speeds the process, and also allows the water to "absorb" more sugar. I don't remember the chemistry of why the sugar doesn't crystallize at room temp, but it doesn't. It is typically made in a 1:1 mixture .. heat a cup of water to boiling, add a cup of sugar, stir until the sugar is dissolved, remove from heat, allow to cool.
It is used in cold drinks like lemonade and iced tea because granulated sugar does not dissolve easily in cold liquids. It requires a lot of mixing, and the saturation point is low, so people who like lots of sugar can't get enough to dissolve to get the taste they desire.
Note #1: this is why Southern-style Sweet Tea is sweetened while the tea is still hot.
Note #2: The sweetness of 1 tsp of sugar is not equivalent to the sweetness of 1 tsp of simple syrup. You will have to find your particular taste point.
Note #3: It can be flavored with just about any flavor extract, such as lemon, orange, peppermint, vanilla, almond, and so on. As I understand it, this is typically how flavored iced teas are made.