Can sleeping with your mouth open cause whiteness in your mouth and on the back of your throat an dryness in your mouth a lot ? thanks.

Sure: But everybody sleeps at least part of the night with the mouth open--that's why your body shuts down saliva flow while sleeping. The "whiteness" can be from mediations, bacterial/fungal infection, or dehydration. Before bed: floss and brush your teeth, use a tongue scraper and rinse with an anti-microbial rinse like listerine. Check with dentist if not gone in two weeks or symptoms develop.

Yes: Man is made to breath thru his nose to humidify the air that is breathed in, besides filtering the air. Breathing in dry air can cause the symptoms you are experiencing. Have you been checked for nasal blockage? Using a humidifer may help.