Well, if you go right back to basics, your yeast need some oxygen at the beginning in order to multiply, but they do not need it to make alcohol out of sugar or honey.
You can either leave your fermentation vessel open like you have done and hope there's enough oxygen coming in from the surface, or you can aerate it really well once at the beginning (and you're right, cold water holds more oxygen than warm or hot water) and use either an open or closed fermenter, or you can aerate your must up to several times a day for the first 1/3 to 1/2 of your fermentation by any of several methods: an air stone with air or oxygen pumped through, vigorous stirring with a sanitized spoon or whisk, or a device that attaches to a drill and spins around in the must, whipping it up.
How long it takes your yeast to appear active can depend on a lot of things, including how much oxygen is in the must when you pitch, how much yeast you pitch, whether you rehydrated properly or whether you dry pitched. But how fast your yeast get started doesn't really have anything to do with the eventual taste of the finished mead. I've had some really slow ferments taste really good and I've had some really fast ones taste really good, I've also had some really slow ones develop off flavours, and I've had some really fast ones taste like rocket fuel.
If the instructions on your wine yeast recommend that you rehydrate in water before pitching, follow the directions, it will give your yeast the best chance they can get. I often get even better results if I make an acclimated starter, just adding sugar can hurt the yeast if you add too much. If the directions on my yeast say to rehydrate for 15 min in 1/4 cup of water, I will do that and then after the 15 minutes, I will add 1/4 cup of my must. Once I see yeast activity, then I'll add 1/2 cup of must, then every hour or so I will add another amount to double the volume, I like to make it up to a litre for a 1-gallon batch and I'll make a gallon of starter for a 5-gallon batch.
If you're working with bread yeast, all bets are off, it seems to be fine with being pitched in dry. One of the experiments I want to do one of these days is make three identical 1-gallon JAO musts but treat the bread yeast differently for each - dry pitch one, rehydrate the next, and make a starter for the last, let them finish, and compare them to see which one tastes best.
Once the yeast have multiplied enough to a sufficient population for your must, they will start making alcohol and don't need the oxygen anymore, but when your yeast are making alcohol, they will be making enough carbon dioxide to displace most if not all of the oxygen that may still have been in the headspace in your fermenter.
Once the fermentation is done (your specific gravity stops moving, there's no further bubbling) the yeast become inactive and will start to sink to the bottom of the fermenter, and this is when you want to rack the clearer stuff to another container so that you leave behind most of the inactive yeast and any of the impurities they may have absorbed through the fermentaiton process so that they don't start releasing it back into the mead as they break down later. You also want to limit any headspace at this point because this is the point where it's becoming more vulnerable to oxidation.
Bread yeast and wine yeast seem to be pretty different in what they need and what they'll tolerate, so if you don't like going through all the steps of racking and watching your headspace, my recommendation is that you stick with the JAO and variations on it using bread yeast.
Why do we take all these little steps with wine yeasts if it doesn't seem to make any difference? Because all the research that vintners and meadmakers have done over the years say that the more of these little steps you take, the better the chance you have of making a good product. However, it's like keeping your body fit and eating right, it only improves your chances of living longer, it doesn't guarantee anything, and smoking only increases your chances of lung cancer, it doesn't guarantee you'll get cancer. There are so many different factors in what can make a mead great or awful that we just like to stack the odds in our favour as much as we're capable of. Failure to perform one individual step may have no impact... or it might make all the difference.
I hope this answers some of your questions.
