Re: Questions About Origin Program
1. Not all the time, but it needs to be running when you're playing.
2. Saves are made locally, but there is also cloud save support. Regardless, Origin will be running while you play.
3. Origin can automatically log in when you start your machine, but it's optional.
Note that "Logged In" in this case doesn't have to mean "online". Origin needs to be online for the initial activation of the game, at which point you can set it to Offline and forget about it, at least until some updates come out.
As for performance, from a personal perspective I don't find it to be an issue. Considering the relative power needed for modern games, I've found the impact of multiple game launchers to be negligible. I can only speak for myself though.
It's curious that you mention DA II, this was integrated into Origin in much the same way as Inquisition is.