Is bronchospasm major problem in asthma?

Partially. Bronchospasm is the most important symptom in asthma - it causes the shortness of breath and all of the discomfort, etc associated with asthma. But the main problem is the underlying inflammation that leads to bronchoconstriction when exposed to allergens, odors and other irritants. The best long-term management of asthma is to treat the underlying inflammation.
Absolutely. Bronchospasm is the problem in asthma. Bronchospasm means that muscle cells in the wall of the breathing tubes (known as a bronchus) contract i.e. Go into spasm. This narrows the internal opening of the tube making it harder to move air in and out. It's like the feeling you get when blowing up balloons. This sensation of increased work of breathing & resultant fatigue is called dyspnea.
Yes... Bronchospasm and airway inflammation causes asthma exacerbations.
One problem. There are two major components in asthma, chronic inflammation of the airways and spasm. One feeds into the other, so if you do not treat both aspects you will not get very good results in your treatment. You can not see inflammation, but you can hear and feel the spasm, the tightness, wheezing and shortness of breath.