I've used plenty of receivers like this with arduino.
The operating current is not really an issue; It is drawing power off of the 5v line, not the digital pins, which is capable of 100mA. In any case they usually have an extra "bat" line that you could plug in to a BEC from one of your speed controllers, and then just connect signal and ground lines to the arduino.
pulseIn works, but not ideally (especially for quadcopters). reading with pulseIn tends to have quite a bit of noise, and runs slow. Those receivers pulse around 50-60Hz, so reading those signals with pulseIn (which has to wait for the signal to finish) will take too long. I say "too long" because in all the open source quad copter programs I've looked at the sensor-read/stabilize update loop needs to run at 100 Hz or more.
What you need to use is the interrupt functionality of the AVR. That way you can detect the time between changes in state of the input pulse without causing the rest of the program to stop. This can get somewhat complicated so I'm not going to write a tutorial here. I recommend you research it yourself and ask new, specific questions here if you have any.
It doesn't magically fix everything for you though; the Uno only has 2 external interrupts. I think most arduino based quads use arduino megas or Leonardos, which have more. I know the APM 2 board has a separate chip converting those 8 single channel PPM signals into a single 8 channel PPM signal that goes into the main controller, despite the fact that it is a mega.
You can look into Aeroquad and Multiwii for inspiration. ArduPilot is open source as well but their code is messier, which makes reading it much more difficult. I don't think it is impossible to use an Uno, but it will be difficult.