For the coding, take a look at MultiWii.com if you want an example of some solid open-source quadcopter coding for Arduino. Many quality commercial boards run this code, but it will work on a stock Arduino board with some sensor modules as well.
Not sure what you mean exactly by "internet/rc" control, but there are many options for wireless control, so it depends on what you plan to do with it. You can use a standard RC transmitter/receiver or even go totally custom and use a digital tranceiver, so you can communicate directly with other digital devices. (PC, Arduino, etc)
The Arduino will be able to catch very short triggers, and interrupts are about the most reliable way of catching short signals or pulses. Instead of having to check constantly if a pin is HIGH/LOW, the Arduino continually monitors it for you, and interrupts the main loop when the state of the pin changes. This leaves the CPU free to do other things in between signals.
Yeah, it can handle counting many thousands of times per second, so handling inputs from sensors on a bike should be no problem, especially if you use interrupts, so it only does work when there is a change of state.
Totally possible on one Arduino, and if speed is a concern, just make sure all of your modules support I2c or SPI, and it will handle it.
You can use interrupts for the hall-effect sensors, so the Arduino doesn't have to keep checking the state, and it will handle it easily enough. You woul have install jets on your bike before worry about overrunning its capabilities.
The amount of sensors is limited by what type of sensor etc, so is kind of impossible to give you a max number, but it can handle multiple I2C devices, so you could run the GPS, SD card and LCD all off I2C.
Basically, you usually want gyro, accellerometer, compass, and barometer sensors which you can get in a single module, then you just need something for wireless control, whether you buy a controller or build one. I used an old XBox controller.
I knew about the USB connection, and was initially looking at the possibility of using a USB host shield to read the data from the controller, but couldn't seem to find any examples or info on whether the Orig. Xbox controllers are supported. The general size of the shields and the extra cost involved were also a factor. 360 controllers seem to be supported, but I'm still unsure about these, so if somebody could clarify, please feel free.
I have always enjoyed taking apart electronics to salvage components and see what makes em tick, and this is a little mod that let me dig into the guts of a relatively straight-forward device: an XBox controller. I have a bunch of old controllers around, along with some Arduino Nano boards, so decided to join another together with an RF module in order to control various devices wirelessly. The controller basically consists of a bunch of potentiometers and buttons arranged in an easy-to-operate kind of way, so this build is fairly simple in theory. As such, the wiring is fairly straight forward, there are just a lot of wires to connect to make use all of the inputs. I chose an APC220 radio module due to its ease of use (UART/TTL), range (1Km), and the fact that it allows communication with computers and Arduino based devices. The settings (frequency, data rate, etc) can also be changed on the fly, so it is fairly versatile. Any Arduino compatible RF module could be used in its place. I believe this would work similarly with a 360 controller, but haven't taken one apart... yet.