You have a few more options with the browser extension, e.g. you can edit the scrobble before it is submitted to Last.fm which can help with title additions, wrong artists or albums, also with badly tagged YouTube videos.
In the meantime Last.fm's general scrobbling feature should be working again, see this staff reply.
No, it will only correct your individual scrobbles to your library. And you have to do this before it gets submitted, i.e. usually after 50% of the track length. After that editing with the browser extension is not possible anymore.
@mylengrave: Right, there was another hiccup with beta scrobbling by Last.fm, but disconnecting / reconnecting it in your Last.fm applications should enable it again.
@Sebasty: There seems to be a problem with the new Spotify desktop version 1.0.80 which has been reported as not scrobbling internally anymore. Users rolling back to older versions could use the internal scrobbling then. This might also be the case with a new Spotify Android app, but I cannot test that.