Do social workers help arrange palliative care?

Yes. Depending on where you are and whether you are in a hospital, skilled nursing facility, home health etc. Palliative care can be arranged by social workers, chaplains, case managers and discharge planners. You'll probably never go wrong by starting with the social worker.
Yes. Social workers can usually direct you to a palliative care team, and most palliative care teams include a social worker.
Yes. In addition many cancer programs have palliative care teams as part of the cancer program. They are a great supplement to the care of the patient and their family. Some of the programs even have guidelines to have a palliative care team evaluate all patients with advanced cancer to see if they can be of help with symptoms and family support.