I was forced (by Skype) to upgrade to the newest version (last Friday). Since then I have no Skype icon in the system tray.
Is this by design? Can it be turned back on?
Skype Preview is a Windows Store app (formerly known as a "Modern", "Immersive", or "Metro" app) and doesn't have (probably can't have) a system tray icon.
Skype for Windows (just called "Skype" as opposed to "Skype Preview" in Start, and says "Desktop app" instead of "Trusted Windows Store app" underneath in search results) is still available for, and still works fine on, Win10 Insider Fast ring. It still displays the icon in the system tray, as you expect it to do.
Win10 preview includes Skype Preview, but you can still use the legacy "Desktop" version too; it doesn't prevent it from working or anything like that! In fact, you can be signed into both at once (though it will try to prevent this), though that's kind of silly.
Unfortunately, if you are using Windows 10 you can't install the old "Legacy" Desktop version any more. The download will direct you to the App Store. It really is a shame.