In power options I don't see any option to shutdown when I press the power ON button, or the option to hibernate when I close the laptop lid. I wonder why, as this is what I use in Windows. I would have definitely preferred the same here. Also there is a slight problem with the headphone connection. It is not recognized if the headphone is connected before the laptop is turned ON. But works fine if I remove it and reconnect.
There are a few questions here, the first one - Yes, there is an option for shutdown with the power button press now, you can set it up using System Settings > Power. The hibernate option there is removed because it do not work well on most of the computers by default. I have managed to achieve this with some system tweaks, if you wish to try it first try if your system can hibernate using the default way - execute into terminal console
systemctl hibernate and if it is successfull go into the last titile of my guide in order to make it work on lid close. Otherwise I do not garantee that it will lead you to a success with your system also but if you wish to try - my guide is on https://elementaryos.stackexchange.com/a/21812/12483 .
About your headphones problem, you better ask another question, because as your title is Hibernate and PowerOff options, only people familiar with the problem there will read your post, but not the people with experience in sound problems and also, StackExchange is a question and answer community and the best way for everyone is different questions to be asked in different posts. It is not a support forum for clients of the elementary OS like you have one for Windows but a community of people who can help each other by sharing their knowledge and experience.