I've always heard that you should wait for water to boil before adding pasta/perogies/vegetables/etc. What is the reason for this? Is it because it reaches boiling point faster? If so, why?
The primary reason is for accuracy and reliability in cooking times. Boiling water is guaranteed (not accounting for altitude) to be at 212 F (100 C). With a set temperature you can then say things like "boil X for 9 minutes" with a very high measure of confidence.
You certainly can cook things in the water as you go, but it's going to be a lot more hit-and-miss. Pasta, for example, will begin to cook before it reaches the boiling point. You would need to measure the temperature, and check the pasta regularly to determine when it was done. You can't give someone an accurate cooking time with this method either. The time it takes for water to boil will vary greatly with stove heat output, pot size & shape, and the amount of water.
When you boil something, you want to efficiently dump heat into it to cook it. If you toss your dumplings or whatever into not yet boiling water, it's the same as just soaking it in cool water before you cook it. You can imagine pre-soaking your dumplings/pierogies would just turn the dough into nasty mush. It's the same thing with tossing them into water that isn't yet hot enough to cook them. Also, once your food is in the water, it's soaking up some of the heat, which means the water will take longer to boil. With nothing in the water, the water gets 100% of the heat from the stove, so it will boil more quickly than if you have the mass of the food floating in it. Consequently, dumping food in "30 seconds before it was going to boil" can actually result in it soaking in sub-boiling water for a lot longer than 30 seconds.
Since water doesn't get any hotter the longer you boil it, you should usually wait for it to boil, but there is usually no benefit to waiting any longer than that. Once it hits the boiling point, the heat is used for the phase change into gas. (The actual boiling) rather than increasing the temperature past the boiling point.
It all depends on what you are cooking: for example if you are making white stock you can (and probably should) start with cold water which you bring to a quick rolling boil and then reduce to a bare simmer for a long time. If you are making soup and you just need to cook the chicken breast to add to the soup, then start with boiling water. If you are cooking legumes, then again start with cold water. For blanching vegetables, cooking rice, or pasta you should always start with boiling water to avoid the problems mentioned earlier but also avoid overcooking (vegetables) or making mushy dishes (pasta, rice). There is no way to achieve the "al dente" feeling of pasta if you start with cold water. For that matter, you also never add cold water say to rice pilaf once it starts cooking. Assume you made a mistake and need to add some water to the pot, it must be boiling hot..
Certainly one reason is the accuracy/reliability, after all physical laws guarantee that water boiling point is around 100 C, with small adjustment for pressure/altitude, salt contents and so on.
Then, another reason is that cooking at different temperatures changes dramatically the end result. As personal anecdotal evidence, cooking pasta at lower temperatures make it feel a lot more gelatinous and "spongy", i.e. horrible.
Think also of the different result you get with boiled meat when you put it in cold water or in boiling water.
When you boil things IN the water AS IT BOILS it is usually VERY easy to determine when it is done as it tends to reach the peak of perfection as soon as the water comes to a full boil. This has been especially true for me when cooking Pastas. Simply add the items you wish to boil to the water before heating, wait for it to come to a complete boil, and then test your pasta in your favorite way! Try it, I bet it works for you!
You might as well put the pasta in from cold because you will achieve an overall shorter waiting time. If you wait for water to boil, then you will inevitably be waiting for longer to cook your food.
I've sat with a stop watch and timed two pots of pasta, one from cold and the other from boiling. Sure, boiling water cooks food faster but waiting for water to boil takes a while and you might as well be burning the candle at both ends by throwing the pasta in first (even cold water softens pasta eventually). You can save yourself around 10 to 15 mins.
The only reason cooking instructions state "from boiling" is for an accurate constant that can be replicated.