You can use replacement rules. The rule depends on what sorts of integrals you have and the way you write them. Suppose that you write the integration region with the syntax Element[{x,y},ImplicitRegion[inequalitites,{x,y}]]. Then you can do this way:

replacementRule = 
  Integrate[f_, Element[{x_, y_}, ImplicitRegion[reg_, {x_, y_}]]] :> 
   DisplayForm@
    InterpretationBox[
     RowBox[{UnderscriptBox[StyleBox["∬", FontSize -> 24], reg], 
       RowBox[{f, RowBox[{"\[DifferentialD]", x}], 
         RowBox[{"\[DifferentialD]", y}]}]}], 
     Integrate[f, Element[{x, y}, ImplicitRegion[reg, {x, y}]]]];
mySampleDoubleIntegral = 
  Integrate[Sin[\[Pi] Sqrt[x^2 + y^2]]/Sqrt[x^2 + y^2], 
   Element[{x, y}, ImplicitRegion[(1 <= x^2 + y^2 <= 4), {x, y}]]];
mySampleDoubleIntegral /. replacementRule

The same integral can be written with other styles, for example with Boole, which may require different rules. You may also want to make rules for Inactive[Integrate].

It is hard to know what you have in mind form a single example.

No, not out of the box. You have to replace \unicode{222c} with {\int\!\!\int}:

StringReplace[expr // TeXForm // ToString, 
 "\\unicode{222c}" -> "{\\int\\!\\!\\int}"]