What determines that a player will or can earn a Safe during the end of mission payday in Payday 2? Are there any weekly limits or anything that changes how and when they are earned?
You get a guaranteed safe drop per week. Which means if it has been more than a week since your last safe drop, your next drop will be a safe.
Other than that, the chance to get a safe is independent from any other drop chance in the game, and is not influenced by perks or difficulty for exemple. This is because the drop chance for a safe is calculated by the server instead of the client.
Beside your "free" drop, the exact probabilities are not known.