How to do basic Civilization V operations with keyboard only in, even just moving

In-game Solutions

There are limited options for playing without a keyboard.

As suggested in the comments, if you want to move a unit without using the mouse at all, you can use a numpad. The 963741 keys give you access to all six directions for movement. This does require you to move the unit one tile at a time

For two issues you raise there are hotkeys.

F5 brings up Social Policies

F6 will bring up the full Technology Tree (for research)

However, you won't actually be able to make any selections at these screens without using the mouse, so that may be of limited value.

Third Party Solution

Expanding upon a previous answer I've posted, 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 (or message bar, as you call it). This will allow you to click "Choose Research", "Choose Production", "Adopt a Policy", etc. The 1750, 860 values correspond to the button's position on a 1920x1080 resolution. You may have to adjust this for your own resolution.

#IfWinActive, Sid Meier's Civilization V (DX11)
z::
 {
   mousegetpos, start_x, start_y
   mouseclick, left, 1750, 860, 1, 0
   mousemove, %start_x%, %start_y%, 0
 }

From a comment it sounds like you want to set distant waypoints for your units to travel to. This is only possible by right-clicking on the tile. The script can be expanded to allow x to act as a right-click for this purpose, as shown below.

x::
{
  mouseclick, right
}