Aww, I'm so sad to read your son is having such problems! it sounds like the medication they have given him is helping, since you've been able to back down to just am & pm with the Albuterol.
So, a few quick facts about Albuterol & the steroids - Albuterol is a short acting medication, that when given in an inhaler or nebulizer, has a local effect on the muscles in the lungs - helping the spasms to relax. Because it is short acting, it will have a rapid effect, and only last a short while, which is why it is sometimes dosed multiple times per day. It can also be used "PRN", which means "as needed", such as for people who have exercise-caused asthma symptoms. It is considered a relatively safe drug, but don't exceed the doses the doctor prescribed (even PRN will have a limit per day).
The steroid medication is given the same way, but it acts as an anti-inflammatory on the lung tissue - helping to relax the inflammation which is causing the breathing spasms. It does NOT have an immediate effect, and is usually given over the period of several days, to make sure that the medication has had the desired effect, & had enough time to prevent a recurrence as soon as the medication is stopped. It is important to take the steroid medication regularly as prescribed, and is not prescribed PRN.
The Allegra your doctor prescribed was most likely to address the nasal & chest congestion that is making the symptoms worse, to get that under control. All 3 medications can then work together to get him on the road to recovery.
Regarding giving your child asthma medications without an asthma diagnosis - there is a difference between a disease state & symptoms, but you can't have the disease without the symptoms. The medications are designed to work on the body in specific ways, to deal with the symptoms. Patients who have asthma disease will need to be on those medications long-term, to prevent the flare-up of symptoms that the disease causes. But respiratory infections & allergies can also cause these symptoms. So, don't be concerned so much with the disease states that the medications are typically used for, because what they are really targeting are the symptoms your son is experiencing.
Whether or not your child does have asthma is not something that can be diagnosed simply by having breathing episodes - because there are many causes for those symptoms that needs to be diagnosed with a physical exam & a lung function test. This is important because even though the treatment for the immediate symptoms may be the same, other treatments for the cause of the symptoms might be different. There is a test where a medication (Methacholine) is given at very small doses, to cause a reaction. Only patients with asthma & COPD will have a reaction (albuterol is given to reverse it as soon as it occurs).
A lot of information, sorry for the book. But it sounded like you were looking for more information on some of the treatments for your son, so I hope some of this is helpful for you, to understand what is being prescribed & why, & maybe give you some background information to have further conversations with your pediatrician. Hope you little one feels better soon! T. :)