If you answer any of these questions in the affirmative, read on. I've spent considerable time figuring out solutions to these problems. One solution, I see most motorcyclists adopt, in the absence of bandanas, is wrap their skulls and faces (covering the mouth and nose) with handkerchiefs - under their helmets. I myself use two scarves every time I ride to my office or any other place which takes a little time reaching - one scarf I wrap around my skull and and the other I wrap around the lower part of my face. Done daily and frequently, it becomes quite a chore - rather tedious.
So here's a better alternative; use balaclavas. What are balaclavas? Very similar to monkey caps used during winters in India. But balaclavas have a different origin (you can read more here). You might've also seen robbers shown in movies wearing such head gear :).

It will not only save you from dust and pollution and sunlight exposure, it's easy to use too - no more knotting bandanas every now and then. It might also help reduce some helmet-related hair problems. Very important to enjoy biking without getting bald. Hair matters :).
Balaclavas come in all sorts of material - stretchable, woollen, cotton, lycra, nylon - depending on where you want to use them.