1) There are a few ways to do this.
Make a MovingToTarget state and an Idle state.
The MovingToTarget state starts the walk animation, and the Idle state starts an Idle animation.
Transition to the MovingToTarget state when the user clicks on the world. Transition to the Idle state when the character reaches the target.
If you want more control, you can make an FSM that just controls the animation of the character. Then you send events to the Animation Graph FSM like "StartWalking", "Stop" etc. The Animation Graph can handle the transitions and even play animation sequences with multiple states.
There's a tutorial on how to set up animation graphs (watch it on YouTube for HD):
2) What GUI system are you using? Unity GUI actions should eat the click... but I'll double check.