Regarding "sea longing":
As noted in another answer, the "sea longing" is actually a longing for Valinor, which simply put is "paradise". When elves first appeared in Middle-Earth, they were invited by Valar ("the gods") to come live with them in Valinor.
Those elves who accepted and started the journey to Valinor are called Eldar and those who didn't are called Avari. Eldar in turn consists of three races: Vanyar, Noldor and Teleri. All of Vanyar and Nolder went to Valinor, but only some of Teleri. The events of Silmarillion focus on most of Noldor returning to Middle-Earth to reclaim the stolen Silmarils.
So those elves who would have "sea longing" would be those belonging to Noldor or Teleri. One of the main reasons is that their family and kin live in Valinor and elves who die also travel there (to the Halls of Mandos) in a form of reincarnation. From the more notable characters of LotR: Elrond, Arwen and Galadriel are of Noldor, Legolas and Celeborn are of Teleri.
The majority of elves in the Mirkwood area are Avari though.
Regarding "lets start with the hobbits first":
After his defeat by the Last Alliance between elves and Men, Sauron had been planning his conquest for several millennium. He was biding his time, waiting for the elves in particular to grow weaker, but also for Valar to lose interest in Middle-Earth.
Sauron had closely been grinding down Gondor over time, slowly but steadily claiming bit by bit of it: Minas Ithil, Ithilien, even Gondor's former capital Osgiliath.
When the One Ring unexpectedly surfaces, events escalate out of Sauron's control. Sauron's plan was to use Saruman to keep Rohan in check and eventually neutralize or defeat that country. When he sees Pippin in the Palantír, he thinks that it is the Ringbearer and that Saruman has found the One Ring. Then as he sends a nazgûl to check, he learns that Saruman has been unexpectedly defeated.
This forces Sauron's hand - he feared that one of his enemies would use the One Ring against him - someone like Gandalf or Aragorn. And also that an alliance between Gondor and Rohan would now happen. So he starts an all-out war, trying to defeat Gondor before Rohan would come to their aid.
But also a war in the north, against dwarves, elves and men there. Sauron had troops around Mirkwood, or could move them there from Mordor unchallenged. But he couldn't easily take an army across the Misty Mountains - it means passing Rohan, or Lórien and Mirkwood. And Sauron couldn't single out one faction at a time in that area, since they all band together as allies (made possible through the events of The Hobbit).
So there's no easy way to get an army to The Shire. Plus everyone regards the hobbits as irrelevant, if they even know of them. The Shire isn't a sensible military objective, since it doesn't even have any military.
As for "hobbits easily subdued", that happens mostly though Saurman's cunning and charisma than out of military force. The hobbits outnumber his gang greatly, as they soon realize when Frodo & company returns to lead them.