Was there ever proposed a theory where the value of Dirac Delta at zero had meaning on itself? Was there ever proposed a theory where $\delta(0)$ has a meaningful value or used in a formal way outside integrals?
Particularly, following Fourier transforms, we can formally obtain
$$\pi\delta(0)=\int_0^\infty dx=\int_{0^+}^\infty \frac1{x^2}dx$$
Were there attempts to treat these integrals as some quantities?
 A: In physics this is a frequent device when we wish to quantize the electromagnetic field. We encounter Dirac delta functions $\delta(\omega-\omega')$ and need to make sense of the case $\omega=\omega'$. The way out is to imagine that the whole of space is enclosed in a cavity, with discrete frequencies $\omega_p=p\Delta$, $p=1,2,\ldots$, and then the Dirac delta becomes a Kronecker delta,
$$\delta(\omega_p-\omega_p')=\Delta^{-1}\delta_{pp'},$$
so $\delta(0)=\Delta^{-1}$. At the end of the calculation we can then send the size of the cavity to infinity, so $\Delta\rightarrow 0$, and if we are calculating physically meaningful quantities this limit will be finite.
A: This works fine in Robinson's framework by choosing for example the Cauchy distribution (in the probability sense, not the Schwartz sense) with an infinitesimal value of the parameter.  The "infinitely tall, infinitely narrow" delta-function this obtained returns the value (of a test function at the point when integrated against it) up to an infinitesimal.  Robinson in his 1966 book proves the existence of delta functions which return the value of the test functions "on the nose" but these are a bit harder to define.
