So I'm with Mark, it's a physics/chemistry thing...which true scientists could explain better I'm sure, but I'll have a go....
The motor and esc can only use the energy that is available to it from it's energy source. An esc and motor that are 'rated' to draw large currents can only do this IF the power source is capable of delivering it. The lipo cells are self limiting, i.e the cell chemistry only allows a given amount of energy to be provided to the load...the advisory numbers that are given with the battery describe the current available over a time period, constant = long time period, and peak = short period, but could also deliver a much higher current over a much shorter time period before being damaged.
Hence the reason that the 'c' rating and capacity can be used to compare a general performance difference from one battery to another.
You will not damage a battery until you drain too much energy from it...which is a power AND time calculation.
The videos you see are of people who have over discharged a battery..by drawing too much current off for too long a time...hence the damage.
A small battery on a fast motor and esc combo will only result in shorter running times at slower speeds when compared to a bigger battery on the same load.
So, for your last question of what motor to choose now that you have the battery sorted...it depends what your looking for, a faster motor will result in more speed for a shorter amount of time, a slower motor will result in slower speeds but for longer run times. And it depends on what your using the car for, racing or bashing, etc etc.
To keep it safe, always run a low voltage cut-off for your battery, this will prevent you taking too much energy from the battery (ie over discharging it) and damaging it, potentially in the way it has been described to you.
I hope this helps
