[Gwaihir the Windlord]

Gandalf knew, or thought, that the entire Quest was hopeless,(Lord of Angmar)

Not totally hopeless, but unlikely. He had faith in it. If it was, as you say, completely without a glimmer of hope, then he would not have attempted it. The point was that the odds were against it, but the great daring of Frodo expedition did manage to pull it off, as Gandalf had hoped they would. Remember the Council of Elrond.

'Despair or folly? For despair is only for those who see the end beyond al doubt. We do not.'

Gandalf goes on to state that he does in fact see the only possibility of the salvation of the West in the proposed mission of Mount Doom.

2. My thoughts are that Gandalf would have stayed west of the Misty Mountains until the Gap of Rohan, and then headed down towards Edoras. (Frodo2968)

>??? It was Gandalf that took the Company into Moria to begin with, so I don't know how you manage to work this out. The course that Aragorn took -- i.e. to Lothlorien -- was exactly what Gandalf had planned. The reasons why the Gap of Rohan was not attempted Gandalf states, when the Company are debating the path they should take.

Hmm, Chieftan, why shouldn't he have gone North of Mordor? The country above the Lithui was fairly safe for a long time. He didn't go East, but the West of Middle-Earth had been well-explored by him. There is no reason why he shouln't have entered this area in his long travels. He knew it well, anyway. I seem to recall though that, to the South, Near Harad was as far as he explored Southwards, if indeed the name Incanus was not given to him in Gondor (in this case it would be a Westron name; certainly it does seem a possible Adunaic word, although in this case 'k' would normally be used).