The resolution question comes up on every product page with more than one variant. The short answer: buy for the background, not for the light. An HDRI lights a scene with its brightness values, and those are the same at every size. The pixels only show when the camera sees the sky directly or in a mirror-like surface.
A 360 degree panorama spreads its width across the full horizon. With a 40 degree horizontal field of view, a typical exterior camera, the frame sees one ninth of the panorama width. From an 8K map that is about 910 pixels of background across the frame; from a 20K map it is about 2,220 pixels. For a full HD render (1920 pixels wide) with the sky in view, 16K or 20K keeps the background sharp. For 4K output (3840 pixels wide), 20K is the minimum when the sky fills a large part of the image, and even then the sky is upscaled roughly 1.7 times. A longer lens makes it worse: at 20 degrees field of view an 8K sky gives only 455 pixels across the frame.
Measured on the files we deliver: an 8K OpenEXR sky (8192 x 4096, 16-bit half float, PIZ compression) is between 105 and 145 MB depending on how much detail the clouds have. The 20K version of the same sky (20000 x 10000) is between 665 and 804 MB. A 10K Radiance .hdr from an sIBL set is 178 MB. Renderers load the whole map into memory, and V-Ray, Corona and Cycles keep a full-float copy, so a 20K map can take 2.4 GB of RAM at render time. On a workstation that is fine; on a render node with many parallel jobs it adds up.
Yes. A 20K .exr resampled to 8K in Photoshop, Affinity Photo, Blender's compositor or ImageMagick keeps the full dynamic range as long as you stay in 32-bit or 16-bit float and do not tonemap. Buying the larger version covers both cases. The other direction does not work: an upscaled 8K sky stays soft.
Not directly. Shadow sharpness comes from the size of the sun in the image and from whether the sun is clipped, see What is an unclipped HDRI?. A 4K unclipped sky gives sharper, more accurate shadows than a 20K sky whose sun was clipped in capture.