Why are birds after Smaug's death important, what happens when the goblins join the battle, and what is "dragon-sickness?"
There are two types of birds that gather after Smaug's death. Firstly there are the carrion birds, who swiftly fly over to where Smaug's corpse is. They are going to feast on his remains. Secondly, there are other birds such as thrushes and finches who return to the mountain. This group of birds represent the return of nature to the mountain which had been held back during Smaug's "reign" of the mountain. The mountain has been described as almost a dead place because Smaug would kill anything that would approach, yet with the death of Smaug, so to comes the return of nature.
Dragon sickness is the disease that the dwarves catch based on their greed and avarice for the dragon's treasure hoard. It is interesting to note how their characters change, in particular Thorin in Ch. 15, who up until now has been a hero type figure, but in this Chapter and his resolution to not share any of the treasure he becomes a distinctly unsympathetic character, and one who we can see has been changed by his greed and his overall desire to find the Arkenstone.
I think that the significance of the birds gathering after Smaug's death is twofold. First it is noted that many of the birds are the kind that eat carrion; they have come to feast on the Dragon. They do not come to the mountain itself, but are off in the distance where the dragon was killed. The other birds: the starlings, thrushes, and finches have come back to signify the rebirth of life there in the mountain, now that the dragon is dead. For as long as Smaug had lived there the mountain was is if dead. Living things could not approach it without fear of being killed by the Dragon hoarding his lifeless jewels. Since the dragon is dead, life can now come to the mountain again.
As for your second question, when the goblins join the battle, the two armies who had been fighting over the dragon's hoard have to unite to fignt the Goblins effectively ending the war between them
Dragon sickness is greed which the dwarves begin to suffer from as soon as they realize Smaug is dead.