You answered your own question.
the focus is off every night time, since it has nothing to "focus" on, while during the day, the camera will refocus within 5-10 seconds.
By using "Schedule" to change to B&W/Night profile BEFORE it gets dark, the camera has something to focus on and will generally stay there until morning. If you allow it to change profiles by itself, by waiting for the day/night sensor to determine its dark out, it will most often be too dark by that time to get a good focus.
And yes, if you keep it in B&W full time, regardless of other profile changes between day/night, it will generally keep a good focus.
B&W itself isnt the problem no, but when it switches from color/day profile to B&W profile at night, the focus will shift, and IF you have something for it to focus on BEFORE it gets too dark, (ie use the Schedule to tell it to switch one hour BEFORE sunset, before it would naturally change itself) it will adjust and keep that focus through the night.
Schedule is what you use to determine what times the day and night profiles are to be used. You adjust it. So on your camera, Schedule (telling the camera to switch to the Night Profile), does indeed affect the focus. Right now I have mine switch at about 7:30 both am and pm
**If it is changing on you AFTER having a good focus on night profile before dark, then you may have it zoomed out too far.. it's been shown that at 85-100% zoom, the camera wont reliably hold a focus day or night.