Why these tutorials don't care about wires when using 10A with 5V to light up strip LEDS?
10 A because 150 LEDS of 60 mA need 9 A.


Correct.


  I pretty sure that my jumper wires will not like 9A.


Correct.


  
  So why tutorials and guides (see links below) don't care about wires when using 10 A with 5 V to ligths up strip LEDs?
  


You'd need to ask them.


  
  What kind of cables should I use to connect both the Arduino and the LEDs?
  


First of all remember that you probably can't run 9 A through the LED strips so you won't be daisy-chaining all the strips. Instead, you will need to look at the datasheet for the strips and work out how many strips you can have in series before exceeding the current rating of the flexible conductors. From that you can figure out how many parallel sets of series connected strips you need. Then you can connect each of those sets directly back to the power supply with a lighter gauge wire.


  
  15 AWG seems big for that, no? I'm not even sure that 15 AWG will fit in the Arduino 5 V pin.
  


I'm from metric land so I'm not familiar with AWG sizes. In any case, you won't be powering the LEDs from the Arduino, so the wires between the power supply and it only need to handle the power required by the Arduino - probably 50 to 100 mA depending what else is connected to it.


  
  Also, I think it may be a good practice to put a capacitor somewhere to protect the Arduino.
  


It's not really going to help. There are some on board already but if the lights cause the 5 V voltage to drop a small capacitor isn't going to help.