Initial rest condition for the linear constant-coefficient differential equations
The notion of initial rest and causality follows only for linear systems (time invariant not required).

For proof of Initial rest being sufficient condition for an LTI system to be causal. Please refer to Problem 1.44 of Allen Oppenheim's book on Signal and Systems. Try and solve it yourself (mostly a homework problem, there is also a solution manual just to let you know in case you are stuck), it doesn't require anything extra other than convolution sum.

For $y(t_o^+)$ and $y(t_o^-)$ relating to $y(t_o)$ post the quote you are looking for clarifications on, it depends on multiple context, generally that would be specifying an initial condition of  a one sided Fourier transform or one sided sequence.