To address your first question: A chemistry teacher in high school once mentioned that since distilled water was free of minerals and many dissolved gasses, it behaved differently in osmotic equilibrium situations than regular ol’ tap water. That is why we always rinsed off our lab glassware with distilled water. Its very lack of dissolved material makes it a great solvent for getting the last remaining “stuff” off of lab ware; because it doesn’t have anything dissolved in solution, it’s like a dry sponge for minerals and solutes. Because of that, and because it’s expensive, I’ve never used distilled water in winemaking but reserve it for lab uses only. Even though largely anecdotal, I know that yeast cells are very vulnerable when they are being rehydrated, and I would want to have the osmotic pressure not be a challenge for them during this time. My fear would be that the distilled water would “suck out” of their delicate membranes necessary minerals like potassium and possibly disrupt the hydration process.
As for spring vs. drinking water; that’s a tougher question to answer. The most important things to make sure are not in your water are chlorine, microbes and any obvious toxins like lead or pesticides. I would bet that both are fine with regards to all of those.
If you use your domestic city water supply it’s probably fine, but I would see if you can get the municipality’s water report that will show you how much chlorine, if any, is in your system. To really make sure you’re controlling your city/domestic water supply for chlorine, I suggest employing a charcoal filter (and sometimes you can buy chlorine-specific filters) housing that can be mounted on faucets or even a garden hose. Sources include Grainger Industrial Supply (every cellarmaster has this catalog on their desk) or even “big box” home and garden supply stores. The water filter approach might be the very best and cheapest in the long run. It’s hard to say, unless you get in touch with the manufacturer, how much chlorine is in purchased “spring” or “drinking” water.
As to your second question: I always like to allow any cork a little time to re-expand back into the bottle. Especially synthetic corks, in my experience, tend to have a harder time re-expanding completely and may indeed leak a little bit.
I will be totally honest with you here and lend you a little of my experience experimenting with many different closure types over the years. I won’t name specific names, but I never have been a fan of synthetic corks. Screwcaps are great; I use them currently in my Garnet Vineyards brands. What I don’t like is the squishy, rubbery, plastic plugs that supposedly work the same as corks by being inserted into the bottleneck. I’ve found over the years that I suffer more premature oxidation, loss of free sulfur dioxide (FSO2) as well as high levels of bottle-to-bottle variation with synthetic corks. Though natural corks can still contribute to TCA (the “corked” defect, perhaps an incidence of 1% or more), I find that the actual instance of that is still less than the combined issues synthetic corks present.