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For the second part of your question, local acceleration does matter for the thrust-to-weight ratio. Isp is exhaust velocity where $g_0$ is divided out by arbitrary convention, your rocket engine performing the same in any gravitational environment. I agree with Christopher James Huff that just formulating the rocket equation in terms of exhaust velocity ...
g0 is indeed an idealized "surface gravity" for Earth. As arbitrary and nonsensical as that is for many applications, you may be better off just directly using the effective exhaust velocity: $$\Delta v=v_{e} \ln \left(\frac{m_{0}}{m_{f}}\right)$$