I'm a newbie in photography, so don't judge me too much, just got the canon 70d and I noticed that when viewing through view finder while in P mode the camera would not focus automatically (I would need to press the shutter button half way down to focus). However while in live view the camera would focus continuously. I have 18-55mm IS STM kit lens with AF enabled.
Live view and viewfinder use different kinds of auto-focus and are different. Just make sure your auto-focus type is set to continuous and make sure you have the focus point selected and over what you want to focus on. You have to keep the button half-pressed, whereas in live view you don't. It is just the way the camera works. To test hold your hand up and focus on it and then move the camera slightly and focus on something in the background. You should be able to see in the viewfinder that the focus has actually moved.
Yes, you can. The easiest way to do so is to enable back button Auto Focus via the menu options and then select AI Servo as your focus mode. As long as you press the AF-ON button the camera will continue to focus on whatever is behind the selected focus points.