One of the unfortunate parts of dementia is how social inhibitions can be lost. Childish behaviour, racist language, inappropriate body scratching in public are just some of the ways this is presented.
My mother didn't pick her nose but went through a long phase of clearing her throat every minute or so. It drove me mad. Then later in her disease she was grinding her teeth constantly, to the point it could be heard 8 feet away. That also drove me mad. There were absolutely no physical reasons for this - it was strictly down to her Alzheimer's.
I hope that once your mother is properly assessed, appropriate medication will be prescribed and it will help ease up these annoying and frustrating behaviours.