Is it possible for chronic sinusitis to cause chronic conjunctivitis?

It's possible. The sinuses are connected to the eyes and ears, so it's not uncommon for them to be affected in chronic sinusitis. An eye md (ophthalmologist) can tell you exactly what the issues are causing the redness and rx appropriate therapy, but it's important to get your sinusitis under control as well. Nasal saline irrigation as well as nasal sprays can help.
Yes. There can be conjunctivitis when you have chronic sinusitis. Often these conditions are associated with dry climates such as yours. Use wetting agents for your eyes and nasal saline rinses for your nose and sinuses.
Yes. The sinuses are very close to the eyes, and a very thin bone separates them. But there are manors easily treated problems that can also cause that. Have an eye doc take a look!