Very scary stuff, I'm sure. But it sounds like you are doing everything to keep your little one safe and healthy.
My understanding from my cousin who has worked as a respiratory therapist and developed adult asthma herself, though, is that the coughing is OK (if uncomfortable for all). It's wheezing, or seeming unable to cough that is the real cause for concern, and that is when to use the emergency inhaler. In fact, I think one of the side effects of albuterol is coughing. Remember, coughing is the lungs trying to work things out. It's not *bad* per se.
As for nighttime, how old is your daughter? My experience for those who have asthma is that it will wake you. If she's not an infant, there's no risk of a spontaneous stop in breathing in the way there is with crib death. (Although, I understand. I still check the 7-year-old and 3-year-old to be sure the chests rise and fall before I drop off). But in reality, kids who are mobile and can get out of bed and communicate have time to do so, even with an asthmatic attack. If she's old enough to get out of bed and go to the bathroom or communicate a bad dream, it really is OK (and healthy for all) for you to sleep, and I'm sure you'll eventually get there. It's healthier for everyone if you sleep, though. If mama ain't rested ... ain't nobody happy!
My whole household has allergies, although no asthma (so far, cross fingers), and we all have terrible hacking coughs right now. Asthma is often linked to allergies, and this is the pollen time of year.
Honestly, I would be reluctant to use the emergency inhaler "to be safe." My advice is to save it for real emergencies. If you use it out of fear, there is a real risk she will become dependent on it to cope day to day. And those medications can cause long-term health problems of their own.