You do know that for Emmet to work, you have to manually set the file type to HTML.
When I do that, with Emmet installed, and type HTML + Tab, I get
I need more than that. I set up my own custom HTML template and open it when I need it.
I don't use Emmet because it used to conflict with other plug-ins I had.
You could disable all your other packages except Emmet and see if it works. (restart Sublime)
If it works, enable your other packages one at a time until Emmet doesn't work. (restart Sublime each time)
Then you can decide whether to keep Emmet or the conflicting package.
If that doesn't work, perhaps you need to re-install Sublime Text.