I never had a need for such app, but I guess if you search for "CCTV" in the store you will find a few. That one looks promising for example.
Just keep in mind that videos quickly grow in volume, so you need a fair amount of free space to keep recording for the whole night.
I doubt it shows noise in dB, but you can try to start some app for measuring dB and record the screen instead (?)
WTF. In the past I updated my tablet and after that it started to discharge within two days (not a week as it used to). The only way to fix it was to clear cache partition. And now they are removing that option.