First of all, I am somewhat puzzled by your recipe. From pizzamaniac's site, I notice that the recipe is for a bread machine, not a stand mixer. But what puzzles me most is the ratio of water to flour (the hydration percent). You are using volume measurements, so to determine the hydration percent, I weighed 3 cups of high-gluten flour (13.8 oz.) and 1 1/4 c. water (10.85 oz.). That yields a hydration percent of almost 73%. Most high-gluten pizza doughs run in the range of 56-65%. With the additional liquid provided by the olive oil, I'm wondering whether you have found it necessary to add additional flour to the bowl of your mixer as you knead the dough or on a work surface on which you knead the dough. It would seem to me that your dough would be very wet.
I will assume for the moment that your recipe is OK. As for your processing techniques, I will offer the following suggestions. Instead of putting everything in the bowl at one time, which is typical of bread machine dough processing, I would first proof the active dry yeast in a small amount of warm water (about a couple of tablespoons of the water called for in the recipe), at around 105 degrees F, for about 10 minutes. I would combine the sugar with the remaining water. Put the flour into the bowl, combine the proofed yeast with the water/sugar mixture, and gradually add to the flour in the bowl. What you want to avoid is placing the salt and sugar in direct physical contact with the yeast, since both can harm the yeast and lead to degraded fermentation of the dough.
Once all the water has been taken up by the flour, add the olive oil and knead that in. If you want to use a rest period (autolyse) at this point or even before adding the olive oil, that is fine but it is not absolutely necessary. This is one of those areas where you can experiment over time and see how you like the results with or without the rest periods. Once the olive oil has been worked into the dough, add the salt and knead that in. Since you are using volume measurements, which can be quite unreliable and imprecise, you may have to add additional water or flour--a teaspoon at a time--to achieve the desired condition of the finished dough.
Once the dough is soft, elastic and smooth with no tears on the outer skin, and neither dry nor wet, but rather tacky, remove the dough ball from the bowl and knead it for a minute or two by hand to be sure that the dough is in proper form. Assuming that it is, I would then divide the dough in half at this time (this is one place where a scale comes in handy to insure equal dough ball weights), since it will allow the dough to cool faster once it goes into the refrigerartor. Lightly coat the two dough balls with olive oil and place in containers. Almost any container (covered) will do. It can be a simple plastic storage zip-type bag or an empty bread bag, which are lightweight and compact, or it can even be a metal container with a lid. Either approach will promote faster cooling of the dough in the refrigerator.
The dough should remain in the refrigerator until you are ready to use it, but I would suggest at least 18-24 hours to be sure that the dough is adequately fermented and will not be tough to shape because of excessive elasticity. The dough should be brought out to room temperature and allowed to warm up for about an hour or two. The temperature of the dough at the time of shaping should be above 50 degrees F (to minimize bubbling problems), but 55-60 degres F may be a better target temperature to use. The 1-2 hour estimated warmup time is an estimate only. The actual time it will take for your dough to be ready to shape will depend on the temperature of the dough when it comes out of the refrigerator and the temperature of your baking area. Obviously, the dough will warm up faster in the summer than in the winter. I wouldn't worry all that much about the moisture on the dough. Some professional pizza operators will let their dough balls (oiled) dry out a bit before covering but I haven't seen that moisture is a big problem in a home environment.
As for your question about water temperature, there are two schools of thought. I favor using cooler water temperatures and try to achieve a finished dough temperature of 80-85 degrees F. Others favor using warmer water. You should try both and decide what works best for you. If you follow the above instructions, in the context of having a reliable recipe, I think you should see an improvement in the quality of your dough. It should not be overly elastic and there should be no need to use a rolling pin. Using a rolling pin may be necessary to get a thin dough for a stromboli, but it should almost never be used for pizza dough (the only exception being for thin crust, cracker-type doughs).
My final recommendation is to consider getting a scale and to get used to weighing the flour and water. Since the recipe you have been using is based on volume rather than weight, it may be necessary to experiment with the amounts of water and flour to get the desired hydration percentage that insures good results.