Watch isn't quartz, you can hear it winding it in the video. To get more than a day out of it wind it more/keep it worn to keep the rotor spinning.
Sounds like the moon cycle and month wheel have been misaligned - a watchmaker in your country can sort that out.
For reference - I bought a genuine perpetual calendar (montblanc) that was misaligned in a similar way, the month changed over on the 24th.
I was the second owner and bought it from a high street dealer who had it fully inspected before sale - no one but me noticed so I wouldn't expect this to be perfect on a chinese rep.
The day, if it's a working indicator will be changed from a recessed pusher on the side of the case most likely.
I don't see a brown spot but did Andrew send you QC photos?
You bought a rep watch - if you expect perfection prepare to be disappointed.