Toria, I cook and bake with alternative milks some. I can't add much about how it keeps, but maybe can help a bit with the other part.
Soy milk comes in varying levels of sweetness and with or without vanilla. The vanilla is almost always sweeter than the non-flavored and cannot be used in much, if any, cooking. A bechamel with vanilla soy milk? Yuk. As noted, it does well in baking - cakes, cookies, etc. I just made a delicious pineapple upside down cake that was gluten and dairy free. The vanilla soy milk isn't even detectable.
Puddings are a bit harder. Not sure if it's a missing chemical (like the casein), but soy puddings do not set as firm; I usually use a bit more thickener. For candies, you need milk. For adding milk to melted chocolate to make ganache or something, you need milk. There is too much water in the soy milk and it will seize the chocolate. I've seen vegan sites say it works, but the one time I tried it, I had a solid seized mess. I am sure there is a trick, but I don't know it.
For cooking, I found using a different non-dairy milk was better. I also don't use milk in cooking all that much, so it was only a problem when doing things like mac and cheese or pastitsio. At the time, I found the best sub was a powdered rice milk that had no sweeteners or flavors in it. I could reconstitute the small bit I needed and not worry about keeping two types of milk in my refrigerator.
That was a few years ago. I now find more places that make completely unsweetened and unflavored soy milk. These work OK; you still can use the oat or hemp milk, too.
If you are seriously considering it, the vegan cooking forums are great to get tips on how to make things work without dairy.