In Dark Souls 2, should the area boss still be alive for PVP to be active in that area? That is to say, should I leave the boss alive so I can get invaded and see summon signs? Or is it still possible to be invaded and see summon signs even if the area boss is dead?
Depending on the area, Regular White summon signs (and their Sun Covenant counterparts) will disappear and won't reappear after an area boss has been defeated. However, Small White summon signs (and their Sun Covenant counterparts) persist as well as Red and Dragon summon signs.
Invasion is also still possible after the area boss is defeated.
Exclusions to this rule include, but are not limited to (possible spoiler):
Drangleic Castle after defeating the Looking Glass Knight and The Forest of Fallen Giants after defeating the Last Giant. This is because the aforementioned area bosses are not the only bosses that appear in that area.
It is possible to see summon signs (Dragon Eye Signs, Red Signs, etc.) and overall be active in a PvP environment even if the boss has been defeated. It was a requirement in previous installments of Souls games (Such as Demon's Souls).
In other words, No, it is not required that the boss is still alive.