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First, in order to show the invariance of the term $\bar{\psi}_L \phi \psi_R$, you have to transform respectively: $\psi_L \to e^{-i \alpha^a(x) \frac{\sigma_a}{2} -i \alpha(x) \frac{Y}{2}} \psi_L$ $\psi_R \to e^{-i \alpha(x) \frac{Y}{2}} \psi_R$ $\phi \to e^{-i \alpha^a(x) \frac{\sigma_a}{2} -i \alpha(x) \frac{Y}{2}} \phi$ (a=1,2,3) so that the ...
Essentially, Higgs self-interactions are allowed because they don't violate any laws or symmetries (i.e. the $\phi^4$ term is gauge invariant, Lorentz invariant etc). Informally, a Lagrangian can (and possibly should) consist of any/all combinations of fields, derivatives of fields etc that respect the symmetries of the theory. In the Standard Model the ...