Yup, it works just fine on flex. I've been eating vegan for a couple of weeks now, with a few exceptions here and there as I haven't been able to fully transition. And actually, I don't think you will be spending fewer points on fat than you were as a vegetarian or omni - if anything I have increased my fat intake (happily) in order to get my points up to the right level. My vegan meals are consistently very low-point so I find I need that extra olive oil or handful of almonds to boost the points.
As far as dairy substitutes, it really depends on what you choose to replace milk with. Unsweetened almond milk is 1 point for a full cup - obviously you will be saving points there if you are changing from ff cow milk. Full-fat soy can be 3 or 4 points depending on the variety you choose. I personally drink low-fat soy milk because my stupid grocer stopped carrying unsweetened almond milk, and that is 2 points for a cup.
Soy cheez is kinda not my thing so I have no advice for you there
Um... with yogurt I spend the same number of points eating soy as I did with cow milk, but I didn't make a habit of eating ff yogurt before. I find it hard to find ff soy yogurt, not that I am looking too hard - I am happy with the little containers of low-fat soy yogurt (3 pts), but be sure to read the label: a lot of soy yogurts are still made with milk cultures to get the bacteria started, and are thus not vegan. Stonyfield Farm Organic, as delicious as it is, is dairy-started and not vegan.
A site you may want to check out is fatfreevegan.com - all of her recipes are low-fat and the recipes she includes on her blog actually have WW points already figured out (th recipes in the archives aren't all so lucky). theppk.com is another good site, as is vegweb.com.
Good luck, keep us posted!