For (2), the answers to Does there exist a continuous function of compact support with Fourier transform outside L^1? may be helpful.

For (3), the answer is yes, although one always needs to rely on some theoretical background. My preferred argument is to note that every bounded linear map $C_0({\bf R}) \to L^1(X,\mu)$ ($X$ and $\mu$ arbitrary) is weakly compact, hence if the two spaces were isomorphic then the identity map on $C_0({\bf R})$ would be weakly compact, hence $C_0({\bf R})$ would be a reflexive Banach space, which it isn't.

I don't know to what extent these remarks answer (1) -- are you looking for an actual reference as to who showed the FT is not surjective?

For (2), the answers to Does there exist a continuous function of compact support with Fourier transform outside L^1? may be helpful.

For (3), the answer is yes, although one always needs to rely on some theoretical background. My preferred argument is to note that every bounded linear map $C_0({\bf R}) \to L^1(X,\mu)$ ($X$ and $\mu$ arbitrary) is weakly compact, hence if the two spaces were isomorphic then the identity map on $C_0({\bf R})$ would be weakly compact, hence $C_0({\bf R})$ would be a reflexive Banach space, which it isn't.

I don't know to what extent these remarks answer (1) -- are you looking for an actual reference as to who showed the FT is not surjective?