Q: Video plays part way and stops

See if there's a Flash update available. Open System Preferences > Flash Player then select the Updates tab then click:  Check Now

If Flash is already up to date, in that same Flash Player pane click select the Advanced tab then click Detete All under Browsing Data and Settings. Quit and relaunch Safari then try a video.

And make sure OS X is up to date by following the instructions here >  How to update OS X and Mac apps

Did you click Delete All in System Preferences > Flash Player > Advanced as I suggested?

Safari - Unsupported third-party add-ons may cause Safari to unexpectedly quit or have performance issues

Safari doesn't load a page or webpage items are missing

Safari/other browsers – Website not loading