: The dwarves were patiently sitting around in Mirkwood awaiting capture: they were probably still groggy from their spider stings, lost and disoriented in the forest, malnourished to boot. Yes, up until then Bilbo had had positive experiences of elves, but this was not Rivendell, these elves had been acting eerie and elusive, and now they'd captured Bilbo's companions. Being careful and using the Ring to his advantage was just plain good sense.

Bilbo was also malnourished and would be tired after fending off the spiders. As to the Elves acting eerie and elusive, this may have been Bilbo's perception while in possesion of the ring.

: Sauron was not actively searching for or trying to draw the Ring to him at the time, being preoccupied by the attack of the original White Council. (Good thing, because Sauron does seem to be able to detect when the Ring is being used, within some range or if in direct line of sight > Amon Hen vs. Sam eavesdropping on Shagrat: he was closer to Sauron but out of line of sight). Would he have been aware of Bilbo's any uses of the Ring within the same forest he inhabited had he not been otherwise occupied?

Gandalf left Thorin & Co. prior to their entry into Mirkwood. We later learn that Gandalf journeyed to the Council and was involved in the attack on Sauron's abode.I don't have the time-line at my fingertips, but given the length of time involved it seems unlikely that Sauron was distracted.

Sauron's ability to perceive the Ring, or its use, is a good topic. In my opinion, Sauron is limited in his ability to percieve the ring. Right distance,right direction,maybe if he's really paying attention, or lucky. I think this lack of perception, more than anything else, is tied to his amount of his spirit within the ring. It would be similar to attaching a large piece of uranium to Geiger counter, and then trying to find a small piece of uranium. As to Sauron drawing the ring to himself, my impression has always been that the One Ring actively sought its true master.

: : : Bilbo also had a stronger will than Sam and Frodo and seemed to be able to withstand its influence for a longer period of time.

Then again, the ability to resist the influence could have been tied to Sauron's strength. At the time of The Hobbit, Sauron was regaining his strength, and had not reach his full power.

: Bilbo never had his will tested to the degree that Frodo did, he was never tempted to use the Ring in such critical situations. We have no basis to judge how he'd have handled the stresses and temptations that Frodo (and Sam) faced.

A critical situation depends on a person's point of view. I'm sure there were situations that Bilbo believed were indeed critical. That aside, I have long held the theory that knowledge of the ring's true nature played a great part in temptation and tests of will. Bilbo did not know the true nature of the ring at the time of The Hobbit, neither did Gandalf. At the time of Elrond's council in Rivendell, Gandalf knows the true nature. He refuses to bear the ring because he would be tempted. This brings up an interesting question. Would Boromir have tried to take the ring from Frodo, if he had not known the ring's true nature?