Is the function $u$ a (tempered) distribution? Let $u : C^{\infty}_0 \rightarrow \mathbb C : \phi \rightarrow \int_{0}^{\infty} e^{-x} \phi(x) $ $dx $ Is $u$ a distribution ? Is it a tempered distribution ?

Yes to both (provided you appropriately change $u$ to act on Schwartz functions when asking if it's tempered). In general, any $L^p$ function $f$ defines a tempered distribution $T_f$ (and hence a distribution), where the action is given by integration, and $e^{-x}$ is clearly $L^p$ on $0,\infty)$. See e.g. [Proof that $L^p$ functions define tempered distributons