Why are you referencing the videos on YouTube? I assume (perhaps wrongly) that you own the content, and that you have access to the source files. Is the issue that you don't want to have a giant folio due to video?
Autoplay won't autostart the video. It will autostart the Overlay, which will then invoke the poster frame from YouTube. Apple requires that video that's streamed in a web overlay have a "play me" button, so you force your reader to tap twice to start any video that's in YouTube unless you autostart the Web Overlay. The desktop Viewer and the iPad Viewer have different rules when it comes to video, and they have different capabilities when it comes to Web Content overlays. I always recommend that designers test on the desktop as a first pass, but that you can't rely on it to prove how your content will behave on a device. The only way to know what your content will do on a device is to test it on a device.
Another issue is that you are using Object>Insert HTML..., which you invoke when you paste the code from YouTube. While really interesting in theory, it introduces some unintended consequences as a result of the Webkit Engine that InDesign used to generate its preview. You see the Flash Player warning because you pasted the embed code. If you saved your code to a text file and saved it as video.html in its own folder, then linked to it as a normal Web Content overlay, you would not see the Flash warning. If you absolutely must continue to use the paste method, then you should select the overlay and set its transparency to 0 in order to hide the poster frame that InDesign generates. You can then put any poster on top of the overlay (such as your Tap to Play iconography (which I like very much, by the way...) and it will disappear when the reader activates the video.
Know that on devices, DPS Viewers rely on the OS for rendering Web content. As a result, if the OS changes (as it did this week in the case of iOS), your Web Viewers may change their behaviors as well.
