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I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function.

Could the constant term of $\mathrm{GL}(2)$-Eisenstein series be used to prove the functional equation of $L$-functions? (standard on $\mathrm{GL}(2)$ or Rankin-Selberg on $\mathrm{GL}(2) \times \mathrm{GL}(2)$ or adjoint/symmetric square on $\mathrm{GL}(2)$)

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I think that the best introduction to the Langlands-Shahidi method still is the following book by Shahidi and Gelbart:

  • Gelbart & Shahidi, "Analytic Properties of Automorphic L-Functions" (1988)

It includes a detailed exposition of the $\mathrm{GL}(2)×\mathrm{GL}(2)$ case, from both the Langlands-Shahidi and the Rankin-Selberg point of view (and the relation between them). Of course, general $\mathrm{GL}(n)×\mathrm{GL}(m)$ and symmetric powers are also treated.

As for your other questions, yes, those examples of L-functions are well covered by this method, see the list here.

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