All characters have your regular standing attack string (AAA, etc.) and dashing attack (dash AS). A half dash is used to bring you closer to your opponent while allowing you to use the regular attack string instead of the dashing attack. This is beneficial for allowing you to juggle a lot longer than normal. Using your example, SAAAA works, but the 5th hit usually misses. You can, however, use S, half dash AAAS half dash AAAS, kicking, etc. because the half dashing closes the distance while allowing your regular attack strings to work.
The timing depends on how far you want your dash to be. You can actually half dash properly with just 3 button inputs. >><, the last > isn't usually necessary if you do it correctly. It IS necessary if you want to have a farther half dash. In the case of a short half dash, just press it as fast as you can. For a longer half dash, you do >>.....<>.
As for shifting, the benefits are similar to a half dash. It allows you to manage your position in relation to your opponent during juggles while allowing for the standard attack strings. It's helpful when your space for comboing is limited. Easiest when you use the numkey pad, just press 339, for example, if you want to shift your opponent from heading east in a juggle to northeast.