This is definitely a system codec issue and not an Opera one. Here is a link to the video file from the first post:
https://video.twimg.com/ext_tw_video/635619016179101696/pu/vid/720x1280/M6wxwYfuQHPX2mfP.mp4
If you right-click, save it and play it with Windows Media Player, I bet it will play audio only.

Sorry, but you're wrong. I saved it and it plays perfectly fine with both windows media player and media player classic. Yet there's a black screen if I'm trying to watch it on twitter using opera.

Same for me. For some videos there's only sound and black screen, and some work completely fine. Win 7 ultimate x64, Opera 32.0.1948.69.