Well first thing you have to remember is how the product cycle works, and its entirely possible for you to buy the biggest and most expensive graphics card on the market, and have it replaced by something faster only 6-8 months later. So its never really worth it to buy more performance than you need, because by the time you need it your card is not gunna perform at the level of newer hardware. So you were wise to keep your 7600 GT till this point. But to answer your points directly...
Having an AMD chip, board, and graphics card doesnt really unlock any "special" performance modes or anything just as having an Nvidia motherboard with an Nvidia gpu would. Basically the reason this is encouraged is because you can get software monitoring and overclocking features that look at everything. So if you have an AMD board and GPU, you can use the ATI OverDrive software
here to control everything from one piece of software instead of 3 or 4 different ones. Also note that running an ATI card with an ATI chipset with onboard video, allows you to run your built in video card in addition to your dedicated card, so you can run 2 or 3 monitors off the onboard and 2 more off your dedicated card, and the same thing would apply to running an nvidia chipset with an nvidia card.
The card you are looking at will be SIGNIFICANTLY faster than your 7600 GT, as you can see from the specs
here, the 4830 is in a whole other league. The biggest thing to note is the fact that the 7600 GT was a fixed shader:vertex processing pipeline, where as newer cards from ATI and Nvidia use a unified shader architecture, meaning that you have a number of shader processors that can process heavy pixel or geometry depending on whats needed. So if you have 640 shader processors on that HD 4830, the graphics card can put most of those towards vertex processing in scenes with heavy geometry or it can dedicated those mostly towards shader processing in pixel heavy scenarios, and it can do this on the fly. So where your fixed architecture (like the 7600gt) might do well in one scenario and very poorly in another, the newer architecture is much more flexible and will scale much better.
The HD 4830 is the card I would suggest if you are looking for a good budget gaming graphics card. If you are gaming at 1680x1050 and are looking for great performance throughout any number of games, its money well spent. The HD 4850 is faster, and can be had for $10-$25 more once rebates are factored in, and if you are looking to get more serious about gaming and spend some more time in there...it'll be a good option.
If you are looking for something cheaper that will be a good performance jump, I would suggest the HD 4670, such as
this one on newegg for $65 ($55 after rebate). It is going to be on par with what the previous generation was pushing in the high end. So this HD 4670 will compete with what you saw from the HD 3850, well ahead of the 7600 GT. If you dont do much gaming, or are gaming under 1680x1050, then this is a great way to save money and get a good improvement. In newer games, where the 7600 GT is struggling to get 20 fps, the 4670 will be seeing more than double that.
So you have a lot to gain from any route, just a matter of how much you wanna spend and what you can really utilize. Your gaming resolution should really be the deciding factor here.