I want to make an RC skateboard that can easily take my weight and move me around. How do I go about deciding the electric power system?
Question 1: Does this mean I cannot go above say 11.1V of LiPo voltage?

No. Look at your spec sheet. Maximum voltage is 19 volts.

"Thus this means that at the motor rating of 1000W and our voltage of 6.2V, the current draw will be 161.2A, which is insanely high!"

Question 2: What is going wrong in the above calculation?

Not a blessed thing. What is wrong is that you're blindly doing calculations and not paying attention to the data sheet. Note that the maximum voltage is 19 volts and the maximum current is 53 amps. Multiply those together and you get 1007 watts. Which, being lower than the rated power of 1190 watt, means that the data sheet is off. At your much lower 6 volts, the motor is incapable of putting out the power you want. Also note that this RPM/voltage combination only applies with no load, and this is not how you want to run the motor. Among other things, this does not accurately calculate the current, which will be much, much lower than your 161 amps, even if the motor could tolerate it.

And this is a good thing. 1000 watts is about 1 1/3 horsepower. Any reasonable linkage to the wheels will produce so much torque that turning the motor on would dump on your butt. 

"therefore the required battery would have to be 108mAh." Nope. That's 108 Ah, 1000 times more than you think.

Question 3: Why is the system mass not coming into the picture??

That's because you are not remotely calculating what you need to calculate. In addition to motor friction, the loads on the motor come from drive train friction, road/wheel friction, aerodynamic drag, and potential energy transfer (which occurs when you go up hill.) System mass only directly affects road/wheel friction and potential energy issues. You need to specify the maximum grade you want to handle without losing speed, make some assumptions about your drive train efficiency, and either calculate or measure the drag produced by a person standing upright at 30 km/hr.

Question 4: How would you design the power system for this project?

I'd use a 60A ESC and a very large battery pack. Then again, I'd start small and try to get a feel for how these things work, especially if I were like you, with no idea of what I was doing.