2 Put in details of local calculations.

Q1) I am going to go for "no". I think it's precisely Tate's thesis that shows that the gamma factor for Riemann zeta can be interpreted as a local factor analogous to the usual local factors at the finite primes. However let me absolutely stress that you do not need to read all the technical details of Tate's thesis to understand the analogue completely. Any function on a local field (including $\mathbf{Q}_p$ and $\mathbf{R}$) has a Fourier transform, which is another such function. If you normalise things in a sane way, then the characteristic function of $\mathbf{Z}_p$ is its own Fourier transform, and there's a standard function on $\mathbf{R}$ that is its own Fourier transform. Now do a certain explicit integral to these functions---the same sort in both cases---this is in Tate's thesis. On the $p$-adic side you get $(1-p^{-s})^{-1}$ and on the real side you get the correct Gamma factor. None of this is a mystery really and is really just scratching the surface of Tate's thesis (the meat of which is the functional equation, not the definition!). I would give details In fact, let's do it now.

So $k$ is either $\mathbf{Q}_p$ or $\mathbf{R}$, and $\mu^*$ is a Haar measure on $k^*$. Now if $f$ a function on $k$, let's define$$\zeta(f,s)=\int_{k^\times}f(t)|t|^s d\mu^*$$(the integral should be over $k^\times$)Let's now compute this integral for some choices of $f$, $k$. If $k=\mathbf{Q}_p$and $f$ is the integrals, characteristic function of $\mathbf{Z}_p$ (which are also easy if you know Haar turns out to beits own Fourier transform) and $\mu^*$ is normalised to make $\mathbf{Z}_p^*$ havemeasure 1, but I'm in then the integral is (breaking up $\mathbf{Z}_p\backslash\{0\}$ into a hurry sum of $p^m\mathbf{Z}_p^\times$ for $m\geq0$) $$\zeta(f,s)=\sum_{m\geq0}p^{-ms}=(1-p^{-s})^{-1}.$$

Now let's try another example: -/let's let $f$ be $e^{-\pi x^2}$, which is its own Fourier transform, let's let Haar measure on $\mathbf{R}^\times$ be $dx/|x|$, and let's compute the integral. It's done on p317 of Cassels-Froehlich---Tate's thesis: we need to compute$$\int_{x\in\mathbf{R}}e^{-\pi x^2}|x|^{s-1} dx$$which is readily checked to be $\pi^{-s/2}\Gamma(s/2)$.

1

I am no expert, but let me give you some guesses as to the answers.

Q1) I am going to go for "no". I think it's precisely Tate's thesis that shows that the gamma factor for Riemann zeta can be interpreted as a local factor analogous to the usual local factors at the finite primes. However let me absolutely stress that you do not need to read all the technical details of Tate's thesis to understand the analogue completely. Any function on a local field (including $\mathbf{Q}_p$ and $\mathbf{R}$) has a Fourier transform, which is another such function. If you normalise things in a sane way, then the characteristic function of $\mathbf{Z}_p$ is its own Fourier transform, and there's a standard function on $\mathbf{R}$ that is its own Fourier transform. Now do a certain explicit integral to these functions---the same sort in both cases---this is in Tate's thesis. On the $p$-adic side you get $(1-p^{-s})^{-1}$ and on the real side you get the correct Gamma factor. None of this is a mystery really and is really just scratching the surface of Tate's thesis (the meat of which is the functional equation, not the definition!). I would give details of the integrals, which are also easy if you know Haar measure, but I'm in a hurry :-/

Q2) I think there is a misunderstanding here (perhaps mine, perhaps yours). The global functional equation reflects Poincare duality in the function field case. I don't really see a local functional equation for $(1-p^{-s})^{-1}$ so I don't really know what you're asking.