No. It's not impossible for food allergies to cause nasal inflammation, but it is unusual. It's much more likely to happen in children. If you're talking about a long term, steady chronic sinus problem, then it's not likely a food allergy. Environmental allergies (pollens, dust, etc) commonly cause or worsen sinus problems.
Yes but not common. A food allergy can cause nasal congestion and runny nose just like an inhalant allergy, but these are not the most frequent symptoms of food allergies. If you eat the food continuously and stay congested several days, it would be possible to get enough obstruction to cause a sinus infection. Often food allergies and inhalant allergies occur together, so they should be treated as a whole program.
No. Food allergy reactions do not cause sinusitis. This is a myth that is perpetuated.