In some cases popcorn ceilings are installed to cover up cracks and uneven ceilings, so probably not the best thing to do, especially in a rental.
For cleaning (dust and cobwebs), I use a feather duster made of synthetic “plumes,” but also a very soft-bristled broom dies the trick. You could also use a vacuum cleaner if the hose is the type that extends (with a wand on the end). Mine has a brush attachment on the end which seems to do the trick.