Is there any way to connect the Nintendo DSi touch lcd to my Pi?
Depends, see below
Can I use the Display serial interface port on my pi?
100% no. Drivers for this port are in the GPU binary blob, and that blob is 100% closed source; we're a very long way from editing that ourselves.
Will it be directly compatible?
depends on the meaning of directly compatible. if you expect to attach some wires, turn on your pi and see a picture, then the answer is no. You'll almost definitely need to roll up your sleeves and do some research, possibly some kernel recompilation (for which tutorials exist, it's not as scary as it sounds). But if you expect to do a few days of research and THEN write some code and THEN attach some wires, then maybe the answer is yes. It depends.
if not, what do I need to download/install/buy to correctly interface the screen with my pi?
depends, see below, but chances are once you get the actual pins connected, you won't need any extra parts. Maybe a few ICs like a tsc2007. Depending on how many pins the screen itself demands, you might need some more logic. again, it depends.
Can i also use the gpio ports for power or some sort?
depends on the power that the screen draws. the 5V pin on GPIO is a direct pass through to the Pi's USB power supply, so whatever current you have left over after the Pi's had its share will be available for parts. For a model B and a 1A supply, I wouldn't draw more than 200mA max from that pin, which is probably plenty for a small screen, but the voltage might be more than the screen is designed to take. The 3v3 pin is regulated to 50mA so the odds of powering a screen from that are a lot lower. You need to know what voltage the screen is designed for.
Bottom line: you're gonna need a lot more information about how the screen works. A fact you'll see from reading this section of the forum is every screen is different (not an exaggeration). One thing I can tell you right now is it's not going to be plug and play. You won't be able to run some wires and get an image in 20 minutes. It's probably going to be on some horrid fine pitch connector that you might not even be able to buy, so if you plan on actually building anything with it, you'll have to start by finding and buying that connector, and breaking out its pins. Once you actually get some wires on it, you need to find out what the wires are supposed to say to the screen.
If you actually want to control the screen (which you almost definitely do), you need to know how. Screens talk in many different ways and I have no clue how to control this one, personally. Start off by searching for anyone who's attempted a project with this screen before, not necessarily on the Pi. The point is you want to know what protocols the screen uses for commands, what controller is backing it, etc. From there you'll have to write the code to make it work. Framebuffer drivers are needed if you want to put X on it, but if you just want a program to blast data to the screen, that's a bit easier (assuming you know the protocol).
The only part of this that I know, for a fact, is possible is the touchscreen. The DS and all its descendants use resistive touchscreens, which are an old technology and trivial to control with an ADC or a dedicated controller chip (of which there are many, often with existing drivers for the Pi to match). But separating the touchscreen from the display underneath might be another matter.