I've researched and read responses to sort order questions but have not been able to find out if LR will automatically save or remember your custom sort order after closing. Does anyone know if each folder can have one saved custom order in the dropdown? Thanks, Mike.
Each folder has its own sorting order. There is a default but once you change it, the folder keeps being sorted in that order until the sort is changed again. This applies to custom orders and presets such as Capture Time, Added Order, etc.
This unfortunately can get confusing since Lightroom shows the contents of folders recursively and the sort order shown is a property of the selected folder. So if you are on a directory sorted by Time and you then select its parent directory which is sorted by Filename, then the order of the images is going to change in addition to showing any images present in other sub-directories.