"Cusp form" means "cuspidal automorphic form" by definition. So yes, $\theta(z;u)$ is an automorphic form. But it is not a form lying in a cuspidal automorphic representation, because it is not a Hecke eigenform. 

Cuspidal automorphic representations are irreducible subspaces of the relevant cuspidal automorphic $L^2$-space, so most cusp forms are linear combinations of forms coming from distinct (usually infinitely many) cuspidal automorphic representations. For non-cuspidal forms the picture is even more complicated: linear combination is replaced by an integral with respect to some spectral measure which includes various (Hecke-)Eisenstein series. The general theory is due to Langlands (spectral decomposition of $L^2$-automorphic forms).