When Frodo lets himself down, it seems the rope has lost its usefulness, yet it conveniently slithers down to them.  Elvish magic?  Coincidence?  What do you all think?

I am sure that Elvish rope making involves what we would call magic, although the Elves probably don't perceive it as such. Unlike Frodo, I trust Sam's ability to tie a good knot.

Speaking of Gollum...

It seems to be Sam who is continually bringing up the topic.  Does this tell us anything about Sam?  Is this particularly significant?

Sam's the caretaker. Frodo deals with the burden of the Ring, Sam deals with real world problems such as food, water and Gollum.

As Sam muses about Gollum and trying to throw him off the trail, we find Frodo trying to reconcile his physical and psychological weariness with his intense desire to press onward and complete the quest.  This is a hard point to reconcile; if he rests he's delaying the Quest, but if he doesn't rest he will weary to exhaustion, making successful completion of the Quest impossible.  In a case like this, what would be the best choice?  Or is Frodo caught in between a rock and a hard place?  (no puns intended!)

Frodo and Sam both need to rest. Perhaps Frodo's so exhausted that he fears that if he stops even for a little while he won't be able to summon the strength of will to start out again. He can't really, of course, go all the way to Mt.Doom with no rest at all.

*******************************************************
Although now long estranged, Man is not wholly lost nor wholly changed.
Dis-graced he may be, yet is not de-throned,
and keeps the rags of lordship once he owned:
Man, Sub-creator, the refracted Light
through whom is splintered from a single White
to many hues, and endlessly combined
in living shpes that move from mind to mind.
Though all the crannies of the world we filled
with Elves and Goblins, though we dared to build
Gods and their houses out of dark and light,
and sowed the seed of dragons--'twas our right
(used or misused). That right has not decayed:
We make still by the law in which we're made!