Stopping and starting when pressing the same button.

It's because the game quickly checks when you're pressing the button. It will activate both conditions within the time you press the key.

I would use the key to toggle a flag. Then when the flag is either on or off, switch the movements.

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User clicks with right button
Soldier is stopped
Soldier -> movement start
Flag 0 is off

User clicks with right button
Soldier is not stopped
Soldier -> movement stop
Flag 0 is off

Then add an action to each: "Turn flag 0 on".

Lastly, create an additional event:

Flag 0 is on: Turn Flag 0 off.

User clicks with right button: toggle flag 0

Flag 0 is ON:
Soldier movement start

Flag 0 is OFF:
Soldier movement stop

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Test position of player. Click all the arrows that point outside the sides of the frame.

The condition now will read "when active leaves the play area" (or something similar to that)

Then stop the player.

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When player leaves the play area: set flag off?

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This way will work if you're not using the current status of the timer for anything.

User clicks with right button
Soldier is stopped
Timer is greater than 0.02
Soldier -> movement start
Set Timer -> 0

User clicks with right button
Soldier is not stopped
Timer is greater than 0.02
Soldier -> movement stop
Set Timer -> 0

I think you could just try adding this to Sumo's solution:

Flag 0 is ON:
+(limit condition) only one action when event loops
Soldier movement start

Flag 0 is OFF:
+(limit condition) only one action when event loops
Soldier movement stop

I guess the soldier doesn't stop at frame borders because it gets constantly "started" while its status is ON

(but switching constantly while he collides could cause the bug as well, then you may want apply additional code for extra safety)