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I am always confused about the technicalities of Dirac Delta.

On one hand, $\int \delta = 1$, is it ok to say that $\delta$ is in $L_1$?

In fact, for any $p$, does $\delta \in L_p$?

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    $\delta$ is not a function but defines a measure ${\mu}_{\delta}(f) = f(0)$ for a suitable class of functions (eg f in Schwarz space). A suggestive notation is ${\mu}_{\delta}(f)={\int}f(x){\delta} (x)dx$ but this is only notation. There is no function $\delta (x)$ and ${\int}{\delta} (x)dx = 1$ is not strictly true. – 2017-01-05

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No, elements of $L^1$ are functions of a real variable. But $\delta$ is not. Mathematicians say $\delta$ is a linear functional on some function space. Physicists seem to treat $\delta$ (and anything else they feel like) as if it were a function. Sometimes it comes out all right. But saying $\delta \in L^1$ is simply wrong.