Yes; mathematically, the functions f[n] and g[n] in that formula are defined for all n. In practice, those signals have finite duration. The convolution is the sum of the product of one signal times a shift of the other, which is interesting only where the signals “overlap”, and this is what conv calculates.
And yes, in general a convolution is not symmetrical.
See this recent thread about the implementation of convolution: Optimizing Direct 1D Convolution Code