Resuming from password refills continues
Beating a boss other than the final boss gives a new password with the ability gained from the boss and all newly found powerups
Continuing refills lives and keeps all found powerups, but the displayed password omits those powerups found since the last boss was defeated
The major problem here is that limited continues doesn't make sense if you can continue to play by inputing your passwords. It's just a waste of time for the player.
The whole point of limited continues is to simulate arcade games where there is no coins in the machine anymore and the player have to quit. For example in Contra or Battletoads, after loosing a couple of times you cannot continue anymore, and if you want to, you have to restart the game from the very begining. This add challenge to those games since you cannot just waste hundreds of lives trying the same level again and again, you have to learn quick.
In any game where passwords are implemented, limited continues do not make any sense.
As for whether the powerup comes in the new password immediately or after defeating a boss, I do not think it's a major issue. Both make sense, the variant which gives the new password only after beting boss might be more challenging and more harsh to the player, but also better as it ensures the system cannot be abused. If the new password comes immediately, the player can just pick each powerup individually, reset and use his new password to continue the game. With the system where only one password is given per level, the player has to actually pick all powerup without getting a game over, like in a RPG dungeon where you have to get all chests between two save points without loosing, and cannot just go from one save point get each chest individually and reset, which would be much easier.