Obligatory sheet-anchor: this all goes back to the powers of the Three, about which we have few firm statements. So I'm going to have to speculate quite a bit here. We do know that all three could ward off the decays of time and postpone the weariness of the world ('Of the Rings of Power').

Nenya is the Ring of Adamant, and is also connected to water (hence its name). We know, or think, that it directly protected Lothlorien, as well as altering the flow of time there.

We know that Sauron and Galadriel directly contended (Galadriel mentions it at the Mirror), but I don't know that he can have the same sort of interaction with Lorien as a whole. Certainly if Dol Guldur came out to attack the Golden Wood, they'd notice a difference, but I don't think it's necessarily true to say Sauron could perceive it from a distance.

Narya is the Ring of Fire, and Cirdan helpfully tells us (well, Gandalf) exactly what it does:

...but in all it will support thee and defend thee from weariness. For this is the Ring of Fire, and herewith, maybe, thou shalt rekindle hearts to the valour of old in a world that grows chill. ('Of the Rings of Power')

So you're exactly right that Narya could help in the battles of the North. Would that help save Gondor? I feel like the time-lag might mean they only get there to interrupt the Orcs' victory feast - though if the Lord of the Nazgul was elsewhere, Minas Tirith could probably stand a long siege, so that's a possible.

Whether Gandalf would surrender Nenya to Eowyn is an interesting question, but on balance I think not. Galadriel said:

Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others. (FotR)

While Gandalf might like to send Narya with the Fellowship, none of the members of the party would be able to wield it effectively. (The same, sadly, goes for giving Vilya to Arwen.)

But Nenya is definitely there, and Galadriel does imply that its powers include hiding. It's also thematically appropriate for water to be a concealment - this goes back to Tuor, who was given a cloak of invisibility by Ulmo. So yes, they probably could get through the Gap unseen.

Without Gandalf, I think you're right that Theoden would be too 'help help, I'm old and feeble' to do much good - but without a month's stay in Lorien, the Fellowship would reach Rohan long before the Battle of the Fords of Isen. Saruman wouldn't yet have openly declared his army of Uruks - and Theodred, Theoden's son, would still be alive.

It is noteworthy that at this time, there was no First Marshal of the Mark - the position of 'supreme commander' of Rohan lay fallow (because of Theoden's weakness and Grima's influence). But the Second and Third Marshals, Theodred and Eomer, showed that they were definitely willing to take the fight to Isengard.

The image I'm getting here is of a War of the Ring made up not of brief, heroic battles, but of a string of sieges. The North lays siege to Dol Guldur. Rohan lays siege to Isengard. Sauron, alerted by Saruman, sends out his own armies and besieges Minas Tirith. And all of this takes place in January (yikes!). The armies of southern Gondor have not yet been mustered - but nor have Sauron's Southrons and Easterlings (and Corsairs) arrived. They'll be trickling in over the next couple of months, and who arrives where and when is likely to be critical.

One thing I'm even surer of now is that the Fellowship has to go through South Gondor. Ithilien is going to fall quickly, and everything east of Anduin and north of Minas Tirith is going to be stuffed with Orcs. The only hope of maintaining secrecy is to get down to what Sauron thinks of as the back-lines, and hope to avoid all the troops coming up from the south.

(Which may not be as hard as I'd supposed; I forgot the month in Lorien when I worked out the timings before. I think, if all goes well, they'll still hit Mordor sometime in March.)

hS