= Solution
Write $\alpha X=X\cup\{\infty_X\}$. Its <open sets> are the open subsets of $X$ together with sets $\alpha X\setminus K$, where $K$ is a <compact subset> of $X$. <Compact subsets> of the <Hausdorff space> $X$ are closed. This fact, and finite unions of <compact sets>, verify the topology axioms, including intersections of an old <open set> with a <neighborhood> of infinity. The inclusion $h_1:X\to\alpha X$ is a <topological embedding> with open image.
An <open cover> of $\alpha X$ has a member containing infinity; its complement is <compact> in $X$ and therefore is covered by finitely many other members. This proves <compactness>. Two <points> of $X$ can be separated in $X$. A <point> $x$ and infinity can be separated by an <open neighborhood> $U$ of $x$ contained in a <compact> <neighborhood> $K$, and $\alpha X\setminus K$. Thus $\alpha X$ is <Hausdorff>. Since $X$ is noncompact, every <neighborhood> of infinity meets $X$, so the embedded copy of $X$ is <dense>. This proves it is a \b[<Hausdorff> compactification].
For a <proper map>, use the standard locally <compact> <Hausdorff> convention: it is <continuous> and inverse images of <compact sets> are <compact>. Define
$$
\boxed{\alpha f(h_1(x))=h_2(f(x)),\qquad\alpha f(\infty_X)=\infty_Y.}
$$
An open subset of $Y$ has open inverse image in $X$, hence in $\alpha X$. A basic <neighborhood> $\alpha Y\setminus K$ of infinity has inverse image $\alpha X\setminus f^{-1}(K)$, open because $f^{-1}(K)$ is <compact>. Hence $\alpha f$ is <continuous> and the diagram commutes. Any <continuous> map from a <compact space> to a <Hausdorff space> is proper: a <compact subset> of the target is closed, its inverse image is closed in the <compact> domain and therefore <compact>. This proves the required properness.
The last requested extension \b[does not exist], even as a <continuous> map. Both sequences $h_1(0,n)$ and $h_1(1,n)$ tend to $\infty_{\mathbb R^2}$, since they eventually leave every <compact subset> of the plane. Their prescribed images would be the constant sequences $h_2(0)$ and $h_2(1)$. <Continuity> would force both distinct <points> to equal $g(\infty_{\mathbb R^2})$, contradicting uniqueness of limits in the <Hausdorff> compactification of the line. The original <product projection> is not proper, as its inverse image of a singleton is an unbounded line.
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