Display of DOP Net was turned off, but simulation is happening automatically. Is there a way to disable this feature? I don't know what I touched, but even though the DOP Net node is not displayed, simulation is occurring...

in your screenshot the popnet1 is still a current node (yellow outline)
so as long as your Viewport is set to Show Current Operator it will still cook it, or if you have Geometry Spreadsheet visible, etc

if you want to temporarily disable cooking of DOP networks regardless you can click on the Brain icon to toggle that