These conditions have technical/subtle interactions. It probably suffices to think about automorphic forms on a Lie group, and not think of the interaction with finite places.
For example, on each K-isotype, the Casimir element is an elliptic operator, but it is not elliptic without specifying the behavior under K. From elliptic regularity, such eigenfunctions are real-analytic.
With K-finite and z-finite, moderate growth implies that all derivatives are of moderate growth, etc. (Borel's little book talks about this for SL(2,R), and his Park City notes recapitulate this. The general argument is similar to SL(2,R), anyway.)
K-finite, z-finite, moderate-growth cuspforms are (up to adjustment of central character) provably of rapid decay, so certainly L^2. This and the previous assertion are part of the "theory of the constant term".
Dropping the moderate growth condition is useful occasionally, as in the "weak Maass form" business.
In a more elementary vein, a choice to require annihilation by a finite-codimension ideal in z is akin to looking at functions on the real line annihilated by a product (D^2-lambda_1)...(D^2-lambda_n), namely, polynomial multiples of exponentials. Certainly not every function on the line is annihilated by such, but the spectral theory (Fourier transform) decomposes a general function into a superposition of certain of these special functions. (Here the analogue of K is {1}.)
Even though it is perhaps not surprising for the applications to repns of reductive Lie groups or reductive adele groups, and of reductive p-adic groups, yes, irreducible unitaries of products $G\times H$ are (completed) tensor products of irreducible unitaries of the factors. However, I think this is not "trivially" true, because it depends on showing that these groups are "type I", meaning that "factor repns" are actually isotypic. Many naturally-occurring groups fail to have this property. This property for p-adic reducitve groups was completely proven (in the supercuspidal case) only as late as 1974, by J. Bernstein. Yes, Harish-Chandra proved type-I-ness for reductive Lie several years earlier.
But, then, granting that, the $L^2$ version of your question is affirmative.
Probably the more general assertion (about spaces of moderate-growth functions) has an affirmative answer, but I would not know how to prove it quickly.
The abstract question about decomposition of irreducible repns of products on larger classes of TVSs is essentially open, I think, even for reductive groups, unless something has happened in the last decade or two.
Edit: for examples like theta kernels, it depends partly on how one chooses to define the things. Even in simpler situations, outside of $L^2$ there is a loss of "semi-simplicity", for example, residues of Eisenstein series are quotients with typically uncomplemented kernels. If the Segal-Shale-Weil/oscillator repn is taken to be the unitary one, then since the groups are type I there is a direct integral decomposition into isotypic components (factor repns), on general principles. There is no a-priori guarantee about multiplicites... although many results are known (Howe-conjecture things, first-occurrence stuff due to Kudla-Rallis and others) about the structure.
Edit-edit: ... which reminds me of a hazard: for example, with real-anisotropic orthogonal groups larger than the symplectic groups in pairings, the trivial repn of the orthogonal group maps (Siegel-Weil) to a copy of a holomorphic discrete series containing Siegel-type Eisenstein series. The point is that the repn is unitary, but the Eisenstein series is not in automorphic $L^2$. Of course there is no paradox, but there is some risk of saying irrelevant/silly things. A similar minor hazard is already present for non-compact arithmetic quotients, since Eisenstein series enter the spectral decomposition "continuously", are not in $L^2$ individually, but of necessity generate unitary repns "abstractly", as would any "generalized eigenfunction" entering a decomposition of a Hilbert space.
Another edit: About Type I... Certainly one direction, that a tensor product of irreducible unitaries of $G,H$ is irreducible unitary, is not difficult. It is the other direction, that an irreducible unitary of $G\times H$ necessarily factors as (completion of) $\pi_1\otimes \pi_2$, that requires something (I'm pretty sure!) In looking at the natural argument to prove such a factorization, at some point one finds that the big repn restricted to $G$ has endomorphism algebra with center consisting only of scalars. If we can conclude that the repn is isotypic (=sum or integral of a single irreducible), then the factorization will succeed. Otherwise, the argument stops, in any case. Alain Robert's book on repns of locally compact groups mentions some not-type-I groups. I do not know enough about them to make a counter-example to a claim that this hypothesis is not necessary. Indeed, conceivably the failure of the proof mechanism does not deny the conclusion...?
Best Answer
$\newcommand{\p}{\mathfrak{p}}$Let $C$ be the class group parametrising the components, say $X = \bigcup_{c\in C}X_c$. Then the Hecke operator $T_\p$ sends component $X_c$ to $X_{c\p}$. In particular, the Hecke operators preserving the components are the $T_\p$ where the class of $\p$ in $C$ is trivial. If $f$ is an eigenform on one component $X_c$, then you can extend it by $0$ on the other components, but that is usually not going to be an eigenform for the whole Hecke algebra. If you look at the automorphic representation generated by this extension, it will in general not be irreducible, but it is going to be a finite sum $\bigoplus_{\chi}\pi\otimes\chi$ where $\pi$ is an automorphic representation and $\chi$ ranges over some subset of the characters of $C$.
Proof of the last statement: Let $\pi$ and $\pi'$ be two irreducible representations occurring in the decomposition of the representation generated by $f$. Then the $T_\p$-eigenvalues of $\bigoplus_{\chi \in C}\pi \otimes \chi$ and $\bigoplus_{\chi \in C}\pi' \otimes \chi$ agree for almost all $\p$ (determined by $f$ if the class of $\p$ is trivial in $C$, and $0$ otherwise), so these representations are isomorphic, and $\pi'$ is one of the $\pi\otimes\chi$. Together with the multiplicity one theorem, this proves the claim.