What is the best way to clip a graphic to a region?

I'm pretty sure this can't be done. As evidence of this I put forward Import[] of .EPS and .PDF with such clipping in it: mathematica imports the shapes unclipped. If there would be some undocumented function to do this clipping, I would assume that Import[] would make use of it.

You can use region functionality (RegionIntersection) to clip primitives, although it will be a bit slow. There are two issues, though.

1. The output if not always a graphics primitive. Sometimes the output of RegionIntersection is a BooleanRegion object, and these objects don't render inside of Graphics. This can be fixed by using BoundaryDiscretizeRegion to convert to a BoundaryMeshRegion that does render inside of Graphics (in M12). If you are using earlier versions of Mathematica, you can use my answer to Make MeshRegion/BoundaryMeshRegion work as a graphics primitive in M11 to enable them to be rendered in earlier versions of Mathematica as well.

2. 2 dimensional multi-primitives (e.g., Polygon) lose the edges where the polygons overlap. This can be fixed by converting multi-primitives to lists of single-primitives.

Here is some code that does this:

ClippedPrimitives[prims_, clip_] := prims /. r_?RegionQ :> clipPrimitives[r, clip]

clipPrimitives[p:Polygon[{__?MatrixQ}], clip_] := ClippedPrimitives[Thread[p], clip]

clipPrimitives[prim_, clip_] := Which[
    RegionDisjoint[clip, prim],
    Nothing,

    RegionWithin[clip, prim],
    prim,

    True,
    Replace[
        RegionIntersection[prim, clip],
        b_BooleanRegion :> BoundaryDiscretizeRegion[b]
    ]
]

For your example (which will be slow because the world multi-polygon consists of 5809 polygons):

Graphics[{
    Green, Rectangle[{-18, 35}, {18, 82}],
    Red, Line[{
        {-10.181224213835492,65.15571149065886},
        {-4.290845998237188,79.14535975270482},
        {14.116585925507536,63.683116936759234},
        {16.32547775635689,43.80309045911497},
        {-0.6093596134882375,35.70382041266731},
        {-16.071602429433796,40.85790135131583},
        {-13.126413321634658,57.79273872116096},
        {-10.181224213835492,65.15571149065886}
    }],
    ClippedPrimitives[
        First @ CountryData["World", "Shape"],
        Polygon[{
            {-10.181224213835492,65.15571149065886},
            {-4.290845998237188,79.14535975270482},
            {14.116585925507536,63.683116936759234},
            {16.32547775635689,43.80309045911497},
            {-0.6093596134882375,35.70382041266731},
            {-16.071602429433796,40.85790135131583},
            {-13.126413321634658,57.79273872116096}
        }]
    ]
}]