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I did read the original question and solved it. I also read and answered Kari's question. The final answer, which solves both issues is:
To remove the share link at the END of the video you must set "rel" to 0, which will remove the related videos and sharing options. (https://developers.google.com/youtube/player_parameters#rel)
You are correct, completely removing the YouTube branding is not possible or legal. However, no one asked to completely remove YouTube's branding. Kari asked to remove the "Watch on YouTube" button from the playback controls, which is accomplished by setting "modestbranding" to 1. (https://developers.google.com/youtube/player_parameters#modestbranding)
The player controls are NOT removed, they are auto-hidden by setting the parameter "autohide" to 1, which will HIDE not remove the controls once the video begins playing. Hovering your mouse over the video will cause the controls to reappear. (https://developers.google.com/youtube/player_parameters#autohide)
You are correct in that wmode does not have to be explicitly set, and using CSS to position the iframe/video is the preferred (and more efficient) method. However, most modern browsers and computers will be able to handle without too much of an issue.
Lastly, "&" is a character. "&" is named/entity character reference, which is actually a shorthand way of saying "&", which is a numerical character reference. All map to the ampersand character (U+0026). (http://www.w3.org/TR/html4/sgml/entities.html)
Most folks use "&" in formatted URLs to make it more readable. However, when passing a URL to another script it is good practice to encode the "&" character to "&".
Happy Holidays to All!
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