I notice that the main chart very often is stuck displaying the whole y-axis range all the way down to y=0, even though the only interesting features are in the top 10% of the chart.
Is there a way to specify a minimum y-axis value, such that the chart never tries to display anything below that value and instead stretches the features above to a wider height?

Sounds like you are plotting 0 values. If 0's in your time series are intentional then skip plotting them.