#1 at home it will be very very close to 1:1 experience.
#2 Whenever you shut down PS4, it goes in standby state. There is no "full shutdown". In that standby state console uses minimal ammount of power, it maintains connection with PSN so that you can command it via smarpthone to dowload games, patches are automatic, and new firmwares.
Additionaly, whenever you are away from home, if your Vita is synced to your console [both need to have same account, both to be "activated" in system settings], then you can remotely turn on your PS4 with Remote Play app on Vita. Quality of the video will depent on UPLOAD SPEED of your PS4 [via lan cable or WiFi to the router], ping between PS4 and you, quality of the WiFi router where you are, and download speed of router where you are.