How can I power a NeoPixel ring and strip through Arduino MKR1000?
The on-board regulator of an Arduino can supply a maximum of 500&nbsp;mA when connected to USB, and up to 1000&nbsp;mA when connected to an external power supply (see Maximum current from Arduino 5V power pins).

If your components are drawing more current than that, then you will need to use an additional separate power supply for the NeoPixel ring - ensuring that the ground lines of the Arduino's supply and the NeoPixel's supply  are connected.

If the Arduino has been smoking then it may already be damaged. Did you see which part was smoking? If it was a capacitor as shown in Is my Arduino dead?, then this answer may help.