I have a Nikon 200 with the AF-S Nikkor 18-200mm and one SB-800 Speedlight. I'm wondering about how much I have to spend additionally if I sell this equipment to get something substantially better--i.e., with better specs and more advanced easier-to -use features that will carry me into the next few years of photography...
That all depends how you define "substantially better", "better specs", and "more advanced easier-to-use".
The D200 was/is a nice camera and plenty of photographers have created lots of visually pleasing and revenue-generating images with that camera.
If you're learning photography, the D200 is quite capable and I'd focus on learning things about shutter speeds, aperture, lighting, and composition before spending more money on gear.
If you understand those things and there are some specific features you desire or usability concerns that you have, provide a bit more information with details and we can probably address them.
I've had a D200 since a few weeks after it went on sale here, and I'm not planning to "upgrade" to anything any time soon. It does what it needs to do and it does it exceedingly well. I was at one point thinking up buying a D3 and relegate the D200 to backup duties, but concluded it wouldn't be a good way to spend my money, so I bought a D70s as a backup instead and used the rest of that money for more lenses.
You sound like you're getting carried away in the upgrade madness that is the DSLR world, best you step back and consider exactly why the D200 doesn't do what you want it to do now, when it did do it yesterday. You'll most likely conclude that it's a perfectly good camera that does everything you need it to do and more, that you've hardly even scratched the surface of its capabilities.
Anything you buy to replace it would have to be in the class of a D3 (or whatever replaces it), and won't be "easier to use". Those "more advanced features" are mostly/merely gimmicks for gearheads, things to put in a marketing brochure or website that have very little practical use. Remember that as long as the camera has a good shutter it's a good camera, and the D200 has an excellent shutter. It also has an excellent light meter and AF system, not required to make a good camera but very handy indeed. And then there's all the other stuff it can do that most people never use. While not as many gimmicks as the "modern" consumer cameras, those gimmicks are just that, baubles to lure the unwary who don't realise they're just gimmicks and not things of any practical use.
You're far better off ditching that 18-200 and getting some quality lenses instead. It's not a bad lens, for a hyperzoom, but then again all hyperzooms are pretty shoddy overall.
A decent lens would be a good start. Any 18-200 superzoom tries to do all things, and ends up doing nothing particularly well. The Nikon 18-200 is a very good 18-200, but even so.
I'd suggest something shortish and fast, a 35 f/1.8 or a 50 f/1.8 will do wonders for action and indoors, available-light photography. The D200 is a very capable camera still. And the fifty can be had for a song.
"More advanced" and "easier to use" are often, but not always in opposition.
I used to have a Nikon D40, and sold it to get a D200. The D40 was easier to use, and the results I got straight out of the camera were more pleasing to me then than the ones I got from the D200. The D200 images needed more post-processing to get to the same level that I appreciated then.
The things I appreciated (and still do!) with the D200 are: in-body AF motor, easier to change options like WB and ISO, and the back AF-ON button.
When I think about it, I wish I had better high-ISO quality and full frame. But I've also realised that the things that hinder my photographic developtmen aren't the D200 or my lenses, it's the lack of time and the lack of commitment to photography.
If you want to "breath new life" into your gear, I'd recommend getting the 35mm f/1.8 DX. It's available for about $200 and is an excellent lens with the D200.
To specifically answer your question, I'd say getting a Nikon D7000 would give you a leg up on features and also ease of use. According to Amazon, you'll have to pay $1,250 for the body, and you might get $400 for the D200 used. The lens and flash don't need to be replaced.