Power sources like batteries and wall warts are voltage sources: they will deliver a fixed voltage, more or less. The current that they supply will be determined by the load you attach to them.
The "Amps" listed on a power supply is not how much current it will put out, it is the maximum current that the manufacturer has decided it can put out safely, without acting funny or overheating. So when you read that the USB port on a computer supplies 5V and 500mA, that means that it will supply 5V and up to 500mA. If you draw more than a 500mA current from your USB port, you would exceed the max current rating of the USB port. USB ports usually have an emergency shut-off mechanism built in, so they would just turn off if you drew more than 500mA out of them. I wouldn't try it though :). Your battery seems to have pretty similar safety features: "Super protection: overshoot, over-discharge, short circuit, over voltage, leakage, and many other protective functions".
If you had a 5V 20 Amp power supply, it will act the same as the USB port power supply, except it will be able to handle putting out much more current. As long as you don't exceed a voltage source's max current rating, the amount of current that a voltage source puts out depends on how much current the load draws, not on the voltage source. For example, a resistor will draw a current in Amps equal to the voltage supplied to it divided by the resistance (I=V/R), via Ohm's Law. For microcontrollers, you can often find their current draw on their datasheet: your MPU 6050 with the Gyro + Accel running has an operating current of 3.8mA when you give it ~3.3V. An Arduino Uno draws about 80mA by itself, though I don't think this includes the 3.3V power regulator on the 3V3 pin. So you're looking at somewhere a little over 84mA (=0.084A) in total. Your battery can supply 2.5A or 1A depending on which of its ports you use, so you are nowhere near exceeding the battery's max current rating, which is a good thing.
There are no fuses built in to the Arduino Uno R3. If you exceed its maximum current limitations, you will mess it up. There's a ruggedino available with more safety features if you're worried about frying your arduino. But as long as you're careful you should be fine with a standard R3.