For Q1, the top version has either typos or mistakes right at the start of the work and the rest seems bad too. To see the mistakes, just compare to the other version. The bottom version seems closer to being correct, but still has a mistake because you have a tau in the result, but the tau=0, and then it is correct. Where did the top version come from?
Still, after the simple correction, the bottom version could do with some better precision in the notation. The functions you are convolving are t u(t) and exp(-3t) u(t), not t and exp(-3t). I know that we often get sloppy because single sided signals are implied, but one can make mistakes and readers might be confused if you are not clear. The starting integral is correct because the limits are chosen to reflect the u(t) functions, but I don't like this approach because it is not mathematically correct.
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)