Yes, this could be asthma. The improvement with the steroid inhaler is consistent with that diagnosis but not specific. And, yes, asthma can cause cough, wheezing or both, not necessarily at the same time. Sounds also like you may have inflammation of your nose, called rhinitis, or sinuses, called sinusitis, and that could be triggering your cough and maybe even the wheezing. You should ask your doctor to check you for this.
We should, however, not assume that asthma is the cause of your cough and wheezing. You should definitely have a chest x-ray and, at some point, preferably when your cough subsides a bit more, you should have pulmonary function tests, both pre and post-bronchodilator.