You're getting the error because Google's webpage requires a secure connection, and Opera can't verify that security (specifically, it says the server's certificate is invalid, meaning it can't confirm that the server it's communicating with is really a Google server, versus an imposter).
As for why you're having this issue, I'm not sure. What kind of network are you connecting through? Home, work, public, etc.? Does it block secure connections?
If you're connecting through a work, school, hotel, or public network, have you gone through all the pages that ask you to accept the terms and conditions and/or pay? If not, the network you're connected to might be trying to intercept your connection with Google in order to show you those pages. Try going to an insecure website like example.com (literally) to let those pages pop up.
I have click to play enabled also. I didn't think to ask about that.
I don't think there is anything--you just click on the greyed out window where the video should be. I haven't seen anything in Opera other than that. OH wait, on youtube flash vidoes, I do see a note to click to run Flash player.