I'm assuming the doctor's diagnosis of asthma is correct.
1. Despite these meds clearing my condition, could I relapse again? Asthma is a condition that is chronic. When a kid has it, sometimes they outgrow it. I don't know of any adults who outgrow it, though the symptoms are sometimes there and sometimes not. If a trigger happens again, like another infection, then yes, it could relapse.
2. How can I be sure that this is asthma and not something else? I would go to a pulmonologist for differential diagnosis. In my personal experience, my allergist was suspecting I had another pulmonary condition in addition to asthma, but said the treatment would be similar. So I never bothered.
3. Will I need to monitor this the rest of my life? It depends on how severe the asthma is. I probably had mild asthma for several years before getting it treated. It got treated once it got more severe. Since they put you on a medicine that sounds like prednisone to me, I would touch base with someone regularly, whether it's an asthma doctor, pulmonology, or internal medicine.
Hope this helps. I'm not a doctor, but have had moderate to severe adult onset asthma for 12 years, a less severe form for 2 years before that, and likely a mild form before that. I wasn't diagnosed, though, until 14 years ago.