= Solution
Uniform continuity means that for every $\varepsilon>0$ there is a single $\delta>0$ such that $d(x,y)<\delta$ implies $|f(x)-f(y)|<\varepsilon$ for all $x,y$. A $d'$-Cauchy <sequence> converges uniformly pointwise to a <bounded function>; passing a uniform-continuity estimate through a sufficiently close member proves that the <limit> is uniformly continuous. Thus $C_{b,u}(X)$ is complete.
If $f\in C_0(\mathbb R^n)$, decay at infinity and compactness of a ball make it bounded. Uniform continuity on a sufficiently large compact ball, together with small values outside it, proves global uniform continuity. Hence $C_0\subset C_{b,u}$.
A <uniform limit> of <functions> vanishing at infinity also vanishes at infinity, so $C_0$ is closed. It is not compact: translate a fixed compactly supported bump of height one so that the supports are disjoint. The resulting <sequence> has pairwise sup distance one and no convergent subsequence.
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