Their involvement is minimal. They built the tunnels that Connor uses to elude the authorities, and Samuel Adams is Achilles' (and later Connor's) primary contact with the organization. They are not involved with the Templars, as previous conspiracy theories would suggest, and seem to have little knowledge of the Assassin/Templar conflict. They have a larger role in the standalone expansion, The Tyranny of King Washington, where they are part of the resistance opposing a corrupted Washington.

No. The Assassins and Templars are both born out of the original war between the Precursors and Humans.

Desmond, Cross, and Dr. Vidic all die. William, Shaun, and Rebecca presumably survive. This game concludes Desmond's story and the Apocalypse 2012 storyline. A new modern day protagonist is introduced in Assassin's Creed IV: Black Flag (2013).

It is a standalone expansion. Unlike previous expansions, it takes place after the main campaign and is played as a separate game. The plot involves Connor being thrust into an illusionary nightmare where Washington has been corrupted by an Apple of Eden, and has crowned himself the King of the American States. We eventually learn that this world, created by the Apple, is actually the product of Washington's nightmare, his fear of what he may become if he were to become too powerful.