Bringing up a new PC with Xamarin and VS 2015 Pro I received the following error:
An unexpected error occurred trying to initialize Android Designer.
Please verify the Android SDK path and the Java Development Kit path
on Tools->Options->Xamarin->Android Settings menu. Please see the logs
for more details.
I had exactly the same problem described (on a new PC, with a new Visual Studio install) and am 99.9% sure the answer is in how updates are applied. Or, more importantly, how they aren't being applied when you think they are.
Go to Tools | Options | Xamarin | Other, and under Xamarin for Visual Studio Updates, Check now for STABLE updates. Make sure you're running at least 22.214.171.1240. You will have to close Visual Studio in the background as the updater is running--it'll warn you. Uninstalling and installing the cross-platform development tools from Visual Studio was not applying this update for me. I had to force the issue.
Now, as mentioned earlier, go to Tools, Android SDK Manager. I noticed my "Android SDK Tools" and "Cross Platform Tools" were out of date and updates were available. I accepted and applied the updates to version 24.4.1. BEAR WITH ME, THIS IS WHERE IT GETS TRICKY--it still didn't work.
Go back to Tools, Android SDK Manager. I immediately noticed that even though I'd just applied the latest update for "Android SDK Tools", (24.4.1), a new version (25.2.2) was now available, along with 17 other updates. I accepted them all.
Bingo. Where I once had the annoying error message and an empty design surface and an empty toolbox with the sad "no usable controls" message, I now have the attached screen snapshot--a design surface where I can see a visual representation of my .axml, and a toolbox from which I can drag and drop controls and they actually show up on my Samsung tablet.