Apparently expressions such as $$ \int \delta (x) f(x)dx = f(0)\tag{1}$$ are widely used in Physics.

After a little discussion in the Math SE, I realized that these expression are absolutely wrong from the mathematical point of view.

My question is: can this lack of rigorousness in Physics affect results (i.e. leading to mistakes or wrong results) or is it OK to use these expressions as long as we keep in mind that the Dirac-delta function is a distribution?

PS: I'm adding an other example of fallacious expressions: the Fourier transform of the $\delta$ function being $1$ ($\hat \delta = 1$). It turns out the Fourier transform of the Dirac is not the function $1$, but the regular distribution associated with the function $1$.