When straining things that clog up the screen, I tend to use a spoon or spatula to move the stuff around, so you can get expose un-obstructed parts of the screen.
Of course, you might end up pushing some of the pulp through the process, but it's generally minimal.
You can then either dump out the pulp as you go (possibly into muslin or cheese cloth, let it cool, then give that part a squeeze), or use the back of a large spoon or a rubber scraper to push the pulp against the strainer to get out some extra liquid. (or use an oven mitt, covered in a plastic bag ... it's messy, but it works).
Of course, if you have a cheap strainer, too much force on it can cause the whole screen to pop out, splashing you and making a quite large mess. (I've since made sure that my strainers weren't basically a screen just held in by pressure to a loop of metal that attached to the handle, but are actually one integral part).
...
If you're looking at buying new equipment, I'd look into getting a chinoise with the proper sized stick (not sure what it's called), which you can use to basically compress all of the pulp into the bottom as you're working.
You won't be adding any ascorbic acid or such -would curdle the milk- but an oil based vitamin such as found inside a vitamin E gel-cap would mix in fine. Would only be as absorbable or bio-available as the original vitamin product.
The idea of fortifying soya comes from the notion that we get a large amount of our nutrients from cows milk and without it, must substitute. The answer in general as to 'how important' is to eat a wide variety of whole foods instead of relying on fortified foods.
Best Answer
It is basically as simple as you say (soak, blend, strain) ... just that you should also boil it (for long enough - cca. 10 minutes) to get rid of the fresh bean (grassy) flavor (mainly caused by oxidation of polyunsaturated oils in the beans), together with some other unpleasant side effects of eating uncooked beans (like the infamous gas-producing effect of some oligosaccharides).
There is a vast number of soy milk preparation instructions online ... this one has always worked really well for me: http://justhungry.com/milking-soy-bean-part-1-soy-milk with all steps and techniques described in detail, and you will also find some continuing posts on how to further turn it into tofu, if you wish so.
Depending on further processing steps and your personal preferences you might choose one recipe/preparation method over another, so I will not recommend any specifics, but in general this should set you on the right track.
p.s. I found that straining/squeezing through cheesecloth/cotton muslin/nut milk bag over a colander is a winning technique ... no need to go any fancier.