How to make screen stay turned on while charging?
This option is still there in developer settings. Those settings have just been hidden since around Android 4.2/4.3. Our developer-options tag-wiki shows you how to enable them:


Go to Settings → About
tap the Build number entry 7 times (hammering the item)
Go back to the Settings main page, Developer Settings should now be visible


In there you'll now find the option to "keep display on".