During Aincrad, how were the players successfully transferred to hospital? Did people outside the game just unplug the equipment on an attempt to pull them out of SAO? And how long did it took to make it to the news and be faced as a serious crisis?

I assume you’re an anime only watcher?  In the novels Kayaba Akihiko point blank tells the players that they’re bodies are being sent to hospitals so they won’t worry about their real life bodies.  He gave authorities a 2 hour leeway on unplugging the nervegears power.  This is the time period Asuna refers to in episode 13 of the anime.

There seems to be a several hour period of time from when the first players realize they can’t log out and the authorities were notified by kayaba, and the time when he summons all players to the starting city and informs them of their fate.  So by the time he tells players they are trapped the authorities have known for hours, and the first 200 people have already died.