Use Convolve with a Dirac Delta function?

Questions
1

In Wolfram Mathematica 11.3 Convolve[1, xDiracDelta[x], x, y] gives no result. But the equivalent function Integrate[xDiracDelta[x], {x, -[Infinity], [Infinity]}] gives the correct result which is 0.

2

Try setting the GenerateConditions option to True to find a conditional expression where the results are valid.

Convolve[x DiracDelta[x], g[x] , x, y, 
 GenerateConditions -> True] (* 0 *)
Convolve[g[x], x DiracDelta[x], x, y, 
 GenerateConditions -> True] (* 0 *)

It appears that the answer is “0” for all conditions and for any function, g[x].

3

Yes, with this option it works properly.