Kate.. My daughter's H grew up until she was about 6-8 months old. But I know this time frame can vary per child. As for the monitering it was up to us as well. A pediatric Dermatologist is the one who prescribed it and then it was for us to monitor. Although we did see the Derm on a weekly basis until our daughter was up to her full dose. We had to take her tempature and heart rate (not obvious to do ugghh..) after every dose until we got to her full doses to make sure her heart rate wasnt affected. But it turned out to be easier than I thought it would be once I got used to it. Now that she's on full dose we see the Derm every 6 weeks.
BTW, if you do get the propanolol, ask the pharmacist to add flavoring to the mix. My daughter loves it.. makes it so much easier!