How can I make an InputField with a proper newline/carriage return?

Ah, figured it out:

Panel @ DynamicModule[{input = ""}, 
  Column[{TextCell["Enter your text here:"], 
    EventHandler[
     InputField[Dynamic[input], String, ContinuousAction -> True, 
      FieldSize -> {40, 7}], 
     "ReturnKeyDown" :> 
      FrontEndExecute[{NotebookWrite[InputNotebook[], "\n", After]}]
     ], Dynamic@InputForm[input]}]]

Here is a widget which I constructed some time ago for my purposes. This is an InputField as well, but it operates on boxes, and as a bonus, the standard syntax highlighting works inside it:

ClearAll[codeInputField];
Attributes[codeInputField] = {HoldFirst}; 
Options[codeInputField] = {
   BaseStyle -> {FontSize -> 14, FontWeight -> Plain, FontFamily -> "Courier"},
   FieldSize -> {20, {2, Infinity}},
   ImageSize -> Automatic, 
   KeyEventActions -> Automatic,
   EnterPressedCustomFunction :> Automatic
};

codeInputField[startingCode_, opts : OptionsPattern[]] :=
  With[{bs = OptionValue[BaseStyle], fsize = OptionValue[FieldSize], 
    acts = OptionValue[KeyEventActions],
    EnterPressedF = OptionValue[EnterPressedCustomFunction], 
    ims = OptionValue[ImageSize]},
   Module[{nb, enterPressed},
     With[{actions = 
       Sequence @@ 
        If[acts === Automatic,
          {},
          Replace[acts, 
           (s_ -> f_) :> (s :> f[Hold[startingCode], Hold[enterPressed], s, nb]),
           {1}]
        ],
        enterF  = If[EnterPressedF === Automatic, Hold, EnterPressedF]
      },
     EventHandler[
       InputField[
          Dynamic[
             startingCode, 
             (nb = InputNotebook[]; startingCode = #) &
          ],
          Boxes, 
          FieldSize -> fsize, ContinuousAction -> True, 
          BaseStyle -> bs, ImageSize -> ims
       ],
       {"ReturnKeyDown" :>
          (
              enterPressed = True;
              enterF[Hold[startingCode], nb];
              FrontEndExecute[NotebookWrite[InputNotebook[], "\n"]];
              If[! ValueQ[nb], nb = InputNotebook[]];
              SetOptions[NotebookSelection[nb], ShowSelection -> True]; 
          ),
          actions
       },
       PassEventsDown -> True
  ]]]];

Here is an example of use:

code = MakeBoxes[Plot[Sin[x], {x, 0, 4 Pi}]];
Dynamic[code]

 RowBox[{"Plot","[",RowBox[{RowBox[{"Sin","[","x","]"}],",",RowBox[{" 
     {",RowBox[{"x",",","0",",",RowBox[{"4"," ","\[Pi]"}]}],"}"}]}],"]"}]

The above dynamic is to monitor the code variable. Now, the input field:

codeInputField[code]

The highlighting is enabled after "Enter" is pressed the first time. You can edit the code to see the highlighting at work and how the code variable is updated.

I also coded an interactive REPL-style cell based on this, which can evaluate code inside it. If there is an interest in it, I could expand to include that as well.

Perhaps something like this?

text = "";
EventHandler[
 InputField[Dynamic@text, String, ContinuousAction -> True],
 {"ReturnKeyDown" :> Paste["\n"]}
 ]