One of Bilbo’s strengths in handling conflicts is that he is adaptable.  One of his weaknesses is that he is stubborn.

Bilbo is one clever hobbit.  The fact that he is able to adapt to new situations is what keeps him alive.  A perfect example of this is his use of the ring when fighting the spiders.  Before he found the ring in Gollum’s cave, he never would have been able to use it or know how to use it.  Yet pretty soon after finding it, he figured out how to use it and was using it against the spiders.

Hobbits are clever at quietness, especially in woods, as I have already told you; also Bilbo had slipped on his ring before he started. That is why the spiders neither saw nor heard him coming. (Ch. 8)

The incident with the giant spiders in Mirkwood shows several of Bilbo’s adaptable traits.  First of all, he is using a sword.  This is where it gets the name Sting.  He not only has a sword, but now knows how to use it pretty effectively as a tool and as a weapon.  He has learned how to use his special hobbit “quietness” in combat situations, and he is able to use the ring.  Put all of these things together, and Bilbo is actually quite formidable.

Bilbo goes into the tree, and cuts the dwarves down where they are hanging in their sacs.  He has most of the dwarves saved before the spiders know what has happened.

His little sword was something new in the way of stings for them. How it darted to and fro! It shone with delight as he stabbed at them. Half a dozen were killed before the rest drew off and left Bombur to Bilbo. (Ch. 8)

Even after he has them out, they can’t really help him fight because they are full of spider poison.  He has to put the ring back on go on the attack as the spiders swarm.  Yet he never wavers, even while he is tired.  He just keeps going at it.

One of the problems, however, is this same insatiable desire.  It can be a threat to Bilbo, and cause conflicts with his friends.  Thorin was hurt when he discovered that Bilbo had stolen the Arkenstone from the dragon's lair and given it to Bard and the elves.  It was the one thing he wanted most of all from the treasure.  It was an old family heirloom.  Bilbo believed that with that stone he could avert a war between the elves, men, and dwarves.  He thought he knew better than Thorin, and decided the stone was his for the taking.  Then it diminished Thorin’s trust in him.

Bilbo does have doubts about hiding the presence of the Arkenstone from Thorin and the other dwarves.  They have been his friends and companions.  He feels guilty not telling them about it, since he was tasked with finding it and knows how important it is.  He also worries about the consequences of revealing that he lied.

[If] he had told them at that moment about the Arkenstone, I don't know. …when the heart of a dwarf, even the most respectable, is wakened by gold and by jewels, he grows suddenly bold, and he may become fierce. (Ch. 13)

As Thorin keeps looking for the Arkensone, Bilbo fears what will happen if he is caught with it. He sneaks out when he is pretending to be on watch, and makes to the camp of the men and elves.

When Bilbo offers Bard the Arkenstone, he and the elvenking are surprised he has it.  Bard asks him how he got it.  The elvenking tells Bilbo that he had better stay with them.

"Bilbo Baggins!" he said. "You are more worthy to wear the armour of elf-princes than many that have looked more comely in it. But I wonder if Thorin Oakenshield will see it so. …I advise you to remain with us…” (Ch. 16)

The elvenking is worried about how Thorin will react when he finds out about what Bilbo did, even though Bilbo insists that he took the Arkenstone fair and square as his share of the treasure.  Gandalf is impressed with Bilbo’s solution though, and his insistence that he will go back.

Bilbo does go back, and Thorin is very angry when he finds out what happened later when he tells them all.  He feels betrayed, and threatens to throw Bilbo off the mountain.  He might have too, if Gandalf hadn’t intervened.  

Bilbo takes an incredible risk here, and one that was not really advisable.  You could say that he chose the lesser of two evils.  Either way, it definitely shows how he is stubborn, and makes his own decisions regardless of other people's feelings, and this causes conflicts with others.  I suppose that is both a strength and a weakness.