The short answer on brushless vs brushed motors is that brushless motors are more power efficient. That means you get more turns of the prop for the amount of amps you consume, which is a major help in what you are doing. The only negative is that the speed controllers cost a bit more, but they will fit just fine into the budget you said you have. Gearbox motors have a built in gear reduction so that a low wattage motor can turn a bigger prop at a lower RPM. In some applications, you get better power efficiency that way since the higher Kv motor is inherently more power efficient than a low one is, and so even with the losses of the gear drive you can still be better than just going with a lower Kv motor.
And if you didn't know, Kv is how many rpm the motor will turn per volt delivered to it. A 500 kv motor fed with 10 volts would turn 5000 rpm unloaded. That's theoretically at least. Actual numbers are always a touch lower.