Yes. It is definitely a possibility. Increased coughing in smokers after quitting for the first 3-4 months is very common. It will go away.
No. I have come across some patients tell me this and there are some anecdotal cases of this happening. The only explanation that I can think of is that the the nicotine in cigarette smoke is a vasoconstrictor and keeps the congestion in the nasal passages at bay. When its effect wears off there may be some nasal congestion. I don't think this long lasting.