No other way ?

[Macalaure]

I don't think there was another way to pass the Ephel Dúath. Frodo and Sam tried to cross the smaller Morgai, but couldn't find a possible way. If it was impossible to pass the Morgai, it must have been at least as impossible to cross the Ephel Dúath. 'Ephel' even means 'fence', and I'm sure those mountains didn't carry that name for no reason. I imagine them like an evil (miniature-) counterpart of the (almost) unpassable Pelóri. I doubt there was a different feasible road but the Black Gate and the Morgul Road/Cirith Ungol. The Ered Lithui sound a bit nicer, but not really that much.

A similar question just popped up in my mind: Why did the fellowship try to pass the Misty Mountains by the pass of Caradhras? I mean, the Misty Mountains are an incredibly long mountain range, so it is quite hard to imagine there was no other pass south of Rivendell, preferably crossing a mountain with a more trustworthy reputation.

PS: What about Fladnag and Odorf?
And wouldn't it have been an incredible showdown for Gandalf and the Witch-King if they would have done a magic song contest at the Black Gate like Finrod and Sauron used to?
He hissed a song of wizardry...

[The Might]

The Misty Mountains are a different story, because of the snow and their height, they were harder to pass. And I think Gandalf wanted to go to Lorien before going further, so Caradhras was the shortest way except Moria perhaps, but we all know what Moria meant.

Now, it is quite clear that not all parts of the Ephel Duath were impassable. When the Host of the West refuses the terms named by the Mouth of Sauron we see Sauron's trap:

Down from the hills on either side of the Morannon poured Orcs innumerable.

Of course that are was easier to pass, since it was close to the valley, but still it shows that it was not impossible. I personally believe it might have been possible to pass the mountains somewhere near Durthang. Now the tower had initially been built there to make sure nothing evil comes into Mordor, however since it was close to the valley it couldn't have been a very steep area. Of course it would have been dangerous to pass there as well, but less then Cirith Ungol I'd say. Somewhere near Durthang they could have found some Orc-armour lying around...and the story goes like in the book from there on. That way there might have been no more army to besiege Minas Tirith...

And as far as the Morgai are concerned...it was not impossible. We see them coming to a ravine formed by the small water they encountered, so there clearly were ways of getting down. But they had other, more important reasons to keep going north. They knew they were being looked for after the incident at the gate with the watchers, so the only way was to keep going. And in some cases it would have been no use to attempt such a feat, since Orcs had raised their camps just below the slopes, and they would have noticed Frodo and Sam.

[Legate of Amon Lanc]

Originally Posted by Macalaure

And wouldn't it have been an incredible showdown for Gandalf and the Witch-King if they would have done a magic song contest at the Black Gate like Finrod and Sauron used to?
He hissed a song of wizardry...

You are right, I'm gonna write it. "Alternative road", only I think, Mouth of Sauron would be more appropriate to counter Gandalf's spells. Then, for the sake of Peter Jackson, Arwen could come and release the Fellowship from the darkness of Durthang...

Originally Posted by Macalaure

I don't think there was another way to pass the Ephel Dúath.

Originally Posted by Macalaure

Why did the fellowship try to pass the Misty Mountains by the pass of Caradhras?

`The questions that you ask, Galdor, are bound together,' I say. Here I think is the answer: as the Fellowship didn't attempt to pass the Misty Mountains any other way than through the pass of Caradhras, so they would probably not choose other way through Ephel Dúath.

Originally Posted by Macalaure

I mean, the Misty Mountains are an incredibly long mountain range, so it is quite hard to imagine there was no other pass south of Rivendell, preferably crossing a mountain with a more trustworthy reputation.

Concerning other passes through Misty Mountains, I can assure you that Caradhras was not the only one - there were many. But they were not so good for use. The Hobbit, Chapter 4: Over Hill and Under Hill, these are the first words of the chapter:

There were many paths that led up into those mountains, and many passes
over them. But most of the paths were cheats and deceptions and led nowhere or
to bad ends; and most of the passes were infested by evil things and dreadful
dangers.

The answer is here. The passes through Misty Mountains were rarely used, no maintenance, you could not tell if the path is even still there, or if it is not infested by Orcs. Although Caradhras might have had bad reputation, it was the best choice, at least for the Fellowship's needs at the time: because the pass was more to the south (which would mean going along the western side - safer, generally - as long as possible, and also not so much snow danger, although as we know everything can turn round), the eastern end of the Redhorn Gate was close to Lórien (where the Fellowship wanted to go), and probably it was the safest way in the meaning that there was lesser possibility of it being guarded by Orcs or whatever - as Gandalf says, the Enemy would not expect them to go that way. Also remember that before the Fellowship departed, scouts were sent all over the land and I think it's safe to assume that Caradhras came out as the best choice after considering all the information gathered.

And the Mountains of Shadow? I think it was more or less the same, and thinking about what you posted brings me again to conclusion that even Gandalf or Aragorn could not have had other options to cross the mountains of Mordor, even if they have known of some passage, they wouldn't risk passing it. There were no scouts who could explore the passes of Mordor, and Gandalf&Aragorn wouldn't know if a band of Orcs had not built a tower right at the end of the pass they wanted to use. So I think now it's finally clear that whatever way the Fellowship wanted to use, Ephel Dúath or Ered Lithui would not be the option even if there were million passes in them.