You can embed a YouTube video into any Showkase page which has a text editor where you can paste HTML code (basically any page type except a Gallery Index Page or a Navigation Link).
I would recommend using a Basic page as other pages (such as About, Contact or Gallery pages) may have set layouts and expect other content to be displayed.
On a YouTube video page, you should see a 'Share' icon immediately below the video itself. If you click this, it should reveal the 'Share' section with 'Share', 'Embed' and 'Email' tabs. Click on the 'Embed' tab and YouTube will present you with the required iframe embedding code.
Just copy this code and paste it into the text editor on your Showkase page. Make sure that you click 'Source' on the text editor's toolbar first to enter Soure Mode (otherwise the embedding code will just be displayed on your Showkase page as plain text).
Showkase has its own video fitting routine so you shouldn't need to worry about changing any dimensions.
Is there anyway to show the starting video frame within a gallery page?
Unfortunately, this is not possible.