Maybe. But I remember that he was able to see the Ring Wraiths' true shape when he wore the ring.

Yes, perhaps. However, the land of Mordor in general and Mt. Doom in particular, was not exactly easy to access. I don't even remember if Gandalf had been there before in person. It was very hard to get to Mordor by yourself. Frodo had to meet Shelob first, and pass through her web. I mean, you wouldn't exactly go there to have a picnic on a Saturday afternoon. It wasn't a holiday destination. You wouldn't be able to sit there in the sunshine, wearing a Hawaiian shirt, and drinking your martini surrounded by beautiful women. Female orcs would be available instead.

I am not saying it would be impossible to teleport there. But it would be a lot harder than going to ghe Shire, or Rohan.

That makes perfect sense. If required to replace Gandalf, Prof. Dumbledore would indeed think of the One Ring as a horcrux.