I have progressed somewhat but still there is a problem.For Q2, why do you feel case 1 is not correct? Please explain. Also, what is the definition of the convolution integral and what are the limits of integration in the definition.
However, let's make an observation about case 1. In every step of case 1, it's clear that we are taking the inverse transform of G(s) which will yield g(t). This can be seen by inspection in every formulation. Clearly, G(s) (1) is G(s), and clearly g(t) * δ(t)= g(t)
You should not have ended up with g(τ) because you integrated over the variable τ. When you integrate over a variable, that variable should disappear from the answer. In this case you will get g(t).
I'm sorry if it looked so. But I did understand what you were trying to point out and tried to correct my mistake, and this was the reason I changed the lower limit from 0 to 0-. But I think I should have tried to address the questions posted by you clearly; unfortunately I was (and am) really time pressed yesterday. Sorry.Unfortunately, you didn't try to answer the questions I posed. Hopefully, you at least thought about those questions.