If you clean and seal these areas once, they're easy to *keep* clean. Sometimes it pays to polish these areas, as factory paint is often less than smooth in out of the way areas; smooth paint retains less dirt. For regular cleaning (after I do the exterior/etc.), I use regular wash solution and a small BHB for most of it, MFs and swabs for the tight spots. I rinse by closing the doors *not quite fully shut* and being careful with the water. Sometimes I'll dampen my drying cloth with a little QD.
When doing someone else's car, where they are really dirty, I use scrap rags/paper towels and a degreasing solvent, or 3M Adhesive Remover. Then an all-in-one type product and finally a good LSP. Then I lube the hinges/etc. properly, with a pinpoint oiler that doesn't get lube all over places you don't need it. Excess lube retains dirt too.
I wouldn't use the EZ brush for this, too much risk of marring the painted areas- and polishing marring out of doorjambs is a pain.
Oh, and yeah, I'd sure clean the drainholes and related areas. Why risk a rust-out? I poke the holes with plastic, not something metallic that might scratch through the paint, not something wood that might break off in the hole.