How do you prevent Xcode 7 Playgrounds from automatically running?

In the lower-left part of the Playground there's a right-facing blue arrow, looking like a "play" button.

Do a long click on this button and you will get two options: "Automatically Run" and "Manually Run".

Choose the latter to put the Playground is manual mode and start/stop its execution yourself by clicking on this button.

You can also trigger the code execution by making a custom shortcut to the "Execute Playground" Editor menu command.

Go to Xcode > Preferences... then go to Key Bindings, scroll down to Editor Menu for Playground and add a new shortcut of your choice by double-clicking in the right column in front of the Execute Playground command.

Actually, you CAN map a shortcut for "Execute Playground" to Cmd+R.

Use macOS keyboard shortcuts, instead of Xcode's keybindings.

Go to System Preferences > Keyboard > Shortcuts,

and in App shortcuts, click on +.

Choose Xcode.app, and add a new shortcut for "Execute Playground" (you have to type precisely this!).

Then click on Add.

Now you can use this shortcut in Xcode.

And don't worry for conflicts (at least up to Xcode 9): Execute Playground is not available for standard Xcode projects, and Run is not active for Playgrounds.