When referring to a piece of artwork, should I use quotes or underlining? Also, should the date go in parentheses after the title of the painting or should it be written out?
It depends on the nature of the artwork. Bolding, italicizing and underlining all mean the same thing--they are no longer limited by what we can do with a typewriter, and the relevant concern is the editorial preference of the publication, style manual, or instructor.
If the artwork is something done on an ambitious scale, like a painting, album, novel or an opera, it gets bolded or italicized (There are few uses for underlining in the computer age). If it is something lesser, like a drawing, a short story, or a song from an opera or an album, it gets quotation marks. A television series gets bolded/italicized, but the name of an individual episode gets quotation marks.
You have some discretion about the date in parentheses. If you refer to the artwork multiple times, mention the date after the first time and don't do it subsequently. If the work was produced very recently, like during the present year, don't use the date at all unless you have a compelling reason to do so.