I want to evaluate the following integral
For given arbitrary function $f$ and Dirac Delta function $\delta$ with Heaviside Theta function $\Theta$, what is the form of
\begin{align} \int_{p} f(p) \delta(p+a) \Theta(p) \end{align}
Mathematica just produces $f(-a) \Theta(-a)$ for some condition $a$. i.e. ,$a\in \mathbb{R}$.
It seems one can safely shift the input of the Heaviside Theta function via the Dirac Delta function. Is this interpretation correct?
It is correct if $a\neq 0$. By definition of the Dirac delta $\delta_{-a}(x) = \delta(x+a)$, for any function $g$ continuous at $x=-a$ $$ \int_{\Bbb R} g(x)\,\delta_{-a}(\mathrm d x) = g(-a). $$ In particular, if $f$ is continuous at $x=-a \neq 0$, then $g(x) = f(x)\,\Theta(x)$ is continuous and so $$ \int_{\Bbb R} g(x)\,\delta_{-a}(\mathrm d x) = f(-a)\,\Theta(-a). $$ In particular, if $a> 0$, $\int_{\Bbb R} g(x)\,\delta_{-a}(\mathrm d x) = 0$ and if $a<0$, $\int_{\Bbb R} g(x)\,\delta_{-a}(\mathrm d x) = f(-a)$.
If $a=0$, then it is unclear what the value of $\Theta(0)$ is ... and in general smooth approximation can bring different results, see also these related questions What is the value of the integral $\int_{-\infty}^a \delta(x - a) dx$ and related integrals? and this one What is the integral of $\int_{-\infty}^{+\infty} H(t)\delta(t)dt$ ($H(t)$ Heaviside step, $\delta(t)$ Dirac delta)?
