By default the hotkeys are somewhat limited. As you know, Enter is used for "Unit Needs Orders" and "Next Turn".
You can use some other hotkeys can be used for specific actions.
F5 brings up Social Policies
F6 will bring up the full Technology Tree (for research)
There is no hotkey for Choose Production
An alternative solution is to use Autohotkey.
The following autohotkey script will, when Civ 5 is running, cause z (which is unused by civ 5) to trigger a click on the "Next Move" button. The 1750, 860 values correspond to the button's position on a 1920x1080 resolution.
#IfWinActive, Sid Meier's Civilization V (DX11)
mousegetpos, start_x, start_y
mouseclick, left, 1750, 860, 1, 0
mousemove, %start_x%, %start_y%, 0