What's an "alpine"?

Charle, about tools, instead of buying them straight ahead, rent some equipment if possible, especially if it´s for a course, so that you can see which crampons you feel better with, which height is best for the ice axes etc. Especially considering you´re taking two courses and then Aconcagua which is not technical. It´s besto to rent/borrow and try some options before chosing something for yourself.

By the time you go the Alps, if you really enjoyed climbing (which you probably will) you may buy stuff there, and you can do so in Chamonix, but at least you´ll know what you need and how to buy, plus, there´s the weight issue on airplanes.

Regarding clothes, do invest in good gloves and socks. I highly recommend Black Diamond Front Pointer GTX gaiters, absolutely crampon proof. For sunglasses, go ahead and buy a category 4 already.

I'm all for the mantra "buy nice or buy twice" but if you have not summit ed a peak in sub freezing temps before.. Rent some gear and take it in stride, the mountains are not going anywhere

Before you buy anything you might want to see how your body does with altitude and cold. Everybody is different. I've comfortably hiked/climbed in -40F, while others cant hardly stand it when the temps get to just 10F no matter how much they spend on gear.
Also, do you really need mountaineering school/training?