Designing a Arduino based circuit to power standard and RGB LEDs
First of all you should learn more about electronics and prototype it on breadboard.
Question 1, according to your link, it's say


  Adding a ~470 ohm resistor between your microcontroller's data pin and
  the data input on the NeoPixels can help prevent spikes on the data
  line that can damage your first pixel. Please add one between your
  micro and NeoPixels! Our NeoPixel rings already have this resistor on
  there


Is that clear?



Question 2
If current consume below 500 mA you can use USB. For battery, use the voltage above 7V and suitable current will fine.



Question 3
See arduino schematic in official web for reference. it simply use linear voltage  regulator then convert voltage to 5V. please read more about power consumption of linear regulator. for short you should use 7 at least  because regulator have drop out voltage. if you use too high voltage on raw regulator might get hot and burn due to power loss.



Question 4
Sure it's safer, If you want to control dimming you should use transistor to drive more current.



Question 5
A lot!!


If you have constant 5V from usb or adapter you should power them through 5V pin instead. to avoid voltage dropout
Do you calculate forward voltage of LED? for example white LED have about 3V forward voltage. when you series them you need at least 6V to power its, also you must include forward voltage to led resistor calculation.
If you have 9 leds and each one consume 20 mA. then you put 8 of them in 4 series of 2 leds. you need 20*4 + 20 = 100 mA 

Please learn more and do some experiment.