How to solve the problem of EventHandler losing focus?

Please send a bug report to Wolfram Research ([email protected]). Until you get a patch for your Mathematica version you could the following code ( I only tested it on Windows, but it should work on Mac and Linux, too, I think ):

    Manipulate[
      EventHandler[
        Grid[{{Pane[#, Alignment -> Center, ImageSize -> 222]& @ 
               TableForm[Map[NumberForm[#, {6, 2}]&, {{x, y}, dir}, {2}], 
               TableHeadings -> {{"{x,y}", "dir"}, {"X", "Y"}}, TableAlignments -> Right], 
               (*Deploy @ *)Graphics[{Thick, Green, PointSize[.02], Point[{x, y}], Line[{{x, y}, {x, y} + dir}]}, 
               ImageSize -> {200, 200}, PlotRange -> {{0, 10}, {0, 10}}, Background -> Gray]
               }}, Spacings -> 1, Alignment -> {Left, Right}, Dividers -> None
        ], {"LeftArrowKeyDown" :> (dir = RotationTransform[.1][dir];
        
        click[])
        , 
        "RightArrowKeyDown" :> (dir = RotationTransform[-.1][dir];
        
        click[])
        , "UpArrowKeyDown" :> (If[0 < x + dir[[1]] < 10,
            x += dir[[1]]
          ];      
          If[0 < y + dir[[2]] < 10,
            y += dir[[2]]
          ];     
          click[]
        ), "DownArrowKeyDown" :> (If[0 < x - dir[[1]] < 10,
            x -= dir[[1]]
          ];
          
          If[0 < y - dir[[2]] < 10,
            y -= dir[[2]]
          ];     
          click[]
        )
      }
    ], {{x, 3.2}, ControlType -> None}, 
       {{y, 4.8}, ControlType -> None}, 
       {{dir, {0, 1}}, ControlType -> None}, 
       Initialization :> (click[] := (
        Needs["JLink`"];   
        JLink`InstallJava[];
        JLink`LoadJavaClass["java.awt.event.InputEvent"];
        robot = JLink`JavaNew["java.awt.Robot"];
        robot @ mousePress[InputEvent`BUTTON1UMASK];
        robot @ mouseRelease[InputEvent`BUTTON1UMASK];
        Pause[.01]
      );
    )
]

Your Manipulate's body (first argument) depends on x, y and dir. If any of them changes the whole body is reevaluated and rendered which means the EventHandler after the arrow key press is a new EventHandler, Graphics is also new so it is not surprising that the selection is lost and not restored sometimes.

You can avoid this by making sure that only the essential part is refreshed. Just put Dynamics here and there. It is called 'nesting Dynamics' and explained in tutorial/AdvancedDynamicFunctionality and various threads here:

   EventHandler[
 Row[{Dynamic@
    TableForm[Map[NumberForm[#, {6, 2}] &, {{x, y}, dir}, {2}], 
     TableHeadings -> {{"{x,y}", "dir"}, {"X", "Y"}}], 
   Graphics[{Thick, Green, PointSize[.02], Point[Dynamic@{x, y}], 
     Line[Dynamic@{{x, y}, {x, y} + dir}]}, ImageSize -> {200, 200}, 
    PlotRange -> {{0, 10}, {0, 10}}, 
    Background -> Gray]}], {"LeftArrowKeyDown" :> (dir = 
     RotationTransform[.1][dir]), 
  "RightArrowKeyDown" :> (dir = RotationTransform[-.1][dir]), 
  "UpArrowKeyDown" :> (If[0 < x + dir[[1]] < 10, x += dir[[1]]];
    If[0 < y + dir[[2]] < 10, y += dir[[2]]]), 
  "DownArrowKeyDown" :> (If[0 < x - dir[[1]] < 10, x -= dir[[1]]];
    If[0 < y - dir[[2]] < 10, y -= dir[[2]]])}]