quantum field theory - Clarification of Path Integral formulation

I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have

$$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N \exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j]).\tag{14.28}$$ What happens when we have the left and right most terms i.e. $$<\Phi_{1}|e^{-i\delta H(t_0)}|0>$$ and $$<0|e^{-i\delta H(t_n)}|\Phi_{n}>~?$$ Another thing that I am confused about, where are we using the fact that the state in the beginning and the end is $|0>$? I see we are using the boundaries on time, but I can't see where we use explicitly the fact that we start and end with vacuum i.e. if we had any $$ where would we have a different term? Also in this case, if we are in the free theory (which I assume is the case as the vacuum state is $|0>$, if we start in the vacuum state won't we always end up in the vacuum state? What can happen in a non-interaction theory that can disturb the system from the vacuum?