Can allergies make you cough!?

Yes... Bronchospasm, which can be due to allergies, can lead to cough. The treatment for a bronchospastic cough is bronchodilator and steroid inhalers so you should see your doctor for evaluation and treatment.
Yes. Allergies are the number one reason for chronic dry cough next to acid reflux. See your doctor to determine which. Treatment is simple.
Allergic cough. The respiratory system from the sinuses to the lungs is pretty limited in its natural, healthy response to irritation of any kind, including allergies or infection: you can cough, wheeze, swell up or make mucus. Coughing is not as common as the others but is possible. Other causes should be considered as well if it is peristent.