Distance Entry: On any shot, do you use the distance of the hole or distance of your tee shot?

On the Shot Entry tab, all distance entries (including tee shot) are distance to the hole. They appear on the table under “hole dis”.

Will GM use my phone to determine the yardage to the hole? Or do I need to use a different device and write it down?

You need to manually enter the distance to the hole. Most golfers know this on almost every shot. The problem with GPS is that it can be quite inaccurate and is unusable for putt distances. Note: you never have to compute how long a shot travels – you only need to enter the distance to the hole.
You can write it down and enter later, but many find it easier to enter while on the course (e.g., after each hole is completed).

If hitting lay up on par five, do you enter the distance to the flag or the distance you are trying to lay up to?

Always enter the distance to the flag.

Is the distance entered the distance he hits the ball or the total ball distance?

The user always inputs ‘distance to the hole’ and never ‘distance the ball travelled’ (so no calculations are ever necessary for the user).