I thought he'd keep running too. But it doesn't actually work that way. At least not from what I'm seeing now. Or rather, perhaps only when going UP. Get the camera angle so that you have to push to the right or left to make link walk straight ahead (relative to his own body of course). Now keep holding in that direction. He will start to run in an arc when the camera changes.

Anyway, I passed the third room! Woooooo! I now also have the Ooccoo and have saved the game (outside) while teleported out from the 4th room.

On the jumping attempt where I finally made it, I though to myself, "this is the one, this time you're finishing this bloody room." And I did!

Of course every other time I thought that I ended up in the fire.