I had similar issues. I also thought about a broken device (bluetooth, wifi, usb-controller). I even bought a new pi…
The main issue was the power.
Since i built in my pi in a original NES-Case i screwed a power connector to the case and soldered a usb-cable to it. then i plugged the cable to the mausberry powerswitch and from there to the pi.
With this setup i had the same issues as you described above.
When using a 2Amp power supply and connecting it directly to the pi i managed to get it up and running stable.
good luck ;)