Videos on Youtube should work, they've always worked for me. But the frame rate is very poor indeed.

The flash support is standard with the opera browser, you shouldn't need to do anything in particular. But you could check that you haven't accidentally switched it off: In the browser menu, select Tools and Plugin-modules, check that both plugins are activated.

Also check that your network connection isn't going through a transparent proxy that strips off flash (just trying to cover all the possibilities here..)