about entropy and random number generator
Yes, this is correct. (up to the typos that "so each bit is occurred at" in 1. should read "so each 128-bit value occurs at", and that the unit "bit" is missing at the end of that sentence; as well as the not-mentioned assumption that every time a 128-bit value is randomly picked, its value is independent of the choices take before). If the state of a cryptographically secure PRG is exactly 128 bit, then it's not allowed to "loose" any entropy during operation to be 128-bit secure, and it has to be seeded by a perfectly uniform random source.

The problem is how to choose the initial state. An attacker should have only a chance of $2^{-128}$ to guess the initial state, as (s)he could otherwise just calculate the output of the PRG (after guessing the initial state) to predict its output. So you have to find a way to pick a uniformly distributed random 128-bit value for the initial value, which is a real problem: you need a random source (that cannot be observed by the attacker) and a way to extract 128 bit entropy.