Sounds like a fun project. One thing you need to remember about this bike is that it's not really a mountain bike, but rather a hybrid, and it's therefore never going to be an offroad machine. So long as that's not what you want from it, you'll be fine.
On to the details. Assuming you don't want to take it offroad, you should probably forget about the suspension fork. Everybody thinks suspension = comfort, but bikes were comfortable for a long time before anybody ever slapped a shock absorber on them. Unless you live in a serious pothole-laden urban hell and like to hop curbs, suspension is for offroad bikes. Putting a suspension fork on this bike will make it seem more cushy, but it will also reduce the pedaling efficiency, engender bad riding position and habits because they won't hurt as much, and cost you money that would be more effective elsewhere. As for the stem, it's called a quill stem and it works with a threaded headset (most suspension forks are for threaded headsets). You don't need to swap it, there are plenty of bars out there that will work with it.
As for what to do, you're probably right on with wanting new brakes, though there's nothing inherently wrong with centerpull cantilevers like what I think you have now. There are plenty of good options out there, if you want to order them and do the work yourself (which you should, it's not hard) you can find good prices anywhere, while a local shop which would do the work for you can steer you toward certain stuff they can get you a better value on. At least for now, I would just lube the levers you have and see how they work, you can probably just stick with them; there isn't much to flat bar levers, if they pivot smoothly and don't have flexy plastic levers there's no reason to replace them on a basic around-town bike.
As for getting the comfort you would try to achieve with a suspension fork, you first want to fit the bike well. This might actually require a new stem to adjust the length and angle, but that shouldn't be a big deal; quill stems are pretty cheap these days. Good fit keeps weight off of the hands, which prevents your wrists from getting hammered by every little bump. I would also find some comfortable grips, a set of bar extensions, and a set of gloves. The grips are obvious, while the extensions give you an alternate hand position, which is key when your hands and wrists start to fatigue or get stiff in one place on a flat bar. The gloves serve as a shock and chafe protector, and while they might seem a little serious for the job, I'll guarantee you'll like them, and you'll thank yourself for them in even the slightest falls.
The other place you're going to be able to buy yourself some comfort is in the tires. This is a balancing act between rolling resistance and cush. Larger volume tires and lower pressure favor the latter, while smaller stuff pumped up higher gives you the former.
Hope that helps.