If my late father and I owned a home together as JTWROS, do the furnishings and belongings convey to me with the house or would they fall under the terms of his Will?

Only real property is controlled by the joint and survivorship deed.  While the real property passes to the joint owner as a matter of law and outside any estate, personal property is not controlled by the deed.  As you suspected, the personal property will pass according to your father's will.

Previous answer of legal counsel is correct, real property passes to you by joint tneancy survivorship, and personal property passes by the will, or intestate administration.