It sounds like his asthma is uncontrolled. We went through a hellish 6 months with my 3 year old with similar issues with his asthma. He was hospitalised once and needed an ambulance twice. Fortunately his consultant now has him on a stronger preventive inhalor which has calmed things down exponentially. He now gets colds and has no chest issues at all.
If you've got already done so go back to your gp and ask about better preventive treatment.
Agree with this poster--the asthma is very much uncontrolled. Both of my girls get QVAR (2 puffs, twice a day). We use a spacer (with a mask attachment) with their inhaler--it is the only way little ones can use an inhaler effectively. Follow these steps: https://www.uofmhealth.org/health-library/ug2819. If you use it incorrectly, he's not really getting his needed doses. (Not blaming you--no one showed me the proper way either!)
ALSO, while it's great you have albuterol puffers, I'd also insist on albuterol for a nebulizer. The nebulizer helps humidify the air he's breathing and will help break the junk up in his lungs. This is HUGE. You can also get his steroids for the nebulizer, in the form of Pulmocort (at least, that's the non-generic name).
Finally, with an asthmatic child, I would absolutely recommend buying a finger Pulse Ox (like this:
https://www.amazon.com/FaceLake-Oxi...qid=1486562683&sr=8-3&keywords=pulse+oximeter). For my girls, a reading of 95 while awake tells me they are having trouble. Of course, anything below that is definitely an issue. While sleeping, oxygen numbers will always be lower than while awake. Another important number on the Pulse Ox is the heart rate--a high heart rate and a low oxygen level shows your child is definitely struggling to breathe.
I also own a stethoscope so that I can listen myself for wheezing. There are tons of YouTube videos that can show you how to recognize a wheeze and how to use a stethoscope.
And one more thought: I know you've mentioned your daughter is having issues as well (not asthma related). Have you had someone come to your house to check for environmental issues (lead, arsenic, mold, etc.)?