I'll preface this by saying I'm not a technician in any way, just a musician with some thoughts. From all that you've described, it might indeed be a faulty unit, but let's not jump to any conclusions and try a few things to guarantee it's not something easily addressable. It might be helpful if you can post a video showing how its setup and what the sound is like.
That definitely does not sound like regular behaviour but it's hard to know if that's an issue with the instrument itself, or an issue with how you've setup your init patch. Just to be super clear, have you done a factory reset and built the init patch that the manual suggests?
From what I've read and heard, Neutrons are known to be finicky when it comes to tuning. I had a Neutron about a year ago and ended up selling it due to tuning instability. But what you're describing sounds much more annoying...almost like a problem with the power source (or the oscillators themselves). Are you using the given power supply or USB? An easy first place to start would be by isolating the power (trying on a different plug that has no other electronics plugged in) and seeing if that behaviour continues.
If a factory reset and power aren't the issues, your best bet is to contact the person you got it from, see if they ever had any similar issues. Otherwise you can always contact Behringer for support.
That sounds exciting! I've been debating setting everything up to be able to run through Ableton as well (currently I run everything dawless) but it feels like a big task. Could you give a rundown of your plan? Would love to hear how you're routing everything