I haven't built very much and have no training in construction but I do have an associates in architectural technology so that has to count for something.
For the drywall I'd recommend greenboard which is moisture and mold resistant. You can buy it at Lowes or Home Depot, here is an example of one brand:
http://www.gp.com/build/product.aspx?pid=6729I can't recommend carpet. I mean it might look totally awesome but you have to vacuum it and if you spill drinks or the roof leaks you are in trouble. Linoleum can get bubbles under it and if water gets under it, it could get messed up too. You could go with an indoor-outdoor carpet which you can lay right over plywood and you don't even need any kind of underlay. Personally I would go with just a plain wood floor you can get very nice plywood and sand it down, it looks pretty good and it's very cheap. I can't find a picture at the moment but there are finish grade plywoods that look like wood panels and wood floors. You may also want to consider a rolled rubber floor, which can absorb impact if you end up fighting or something.
You'll notice I keep coming back to the subject of water. I think for a treehouse you really need to plan for getting water everywhere. It's not like a normal structure, there's actually holes open to the outside (I'm thinking your elevator and maybe gaps for your windows depending on how you make them). Roofs leak. Things break.
When drywall gets wet it molds and eventually crumbles. Wood does alright with water. Carpet molds. Any adhesives for your floor will break down from water. Certain metals will rust. Batt insulation will mold. If you build the walls right you should be fine there. The greenboard drywall will be able to handle water balloons, humid days, etc, just don't soak it water. The big issues are water blowing in through doors and windows in driving rain, and your roof leaking, especially if the tree physically punches through a part of your roof. If you do have a tree going through your roof, I would seriously consider building your tree house somewhere different. As an example, my cousin ended up building a house on stilts next to the tree and using the original tree house as a deck. It's actually pretty cool.
Onto actually building the roof. Fortunately your roof is very small. If you want a peaked roof you could put shingles up. You could also use corrugated metal or plastic which would be pretty easy and very cheap. You could do a flat roof and then you'd be able to climb up on the roof too from an external ladder. If it snows where you live be careful with a flat roof though.
There's a lot of materials you can use for everything and you don't have to necessarily be restricted to asphalt shingles and vinyl siding because houses use it. You could go legit and build a house with asphalt shingles, 4 inches of R-19 batt insulation, vinyl siding, etc. Personally I'd skip out on some of the other materials and use it to buy more wood for other cool stuff, or save up for a zip line or something. As an example, I'd probably skip insulation and just turn the heater/ac up. I mean, your parents are paying the electric bill right?
And if you're going to run the cable and electric to it, don't forget to run out a line for a wireless router so you can have wifi too.
If you need any help with choosing materials, figuring out how to install it, etc, I can probably help. I've also got copies of AutoCAD and AutoCAD Architecture so feel free to send me any drawings, I can mark them up and do pretty much whatever else you want (even turn it into a Blockland map, haha), although I'd leave all the design work to you.
