anemones do have to be fed. Photosynthesis of symbiotic zoox doesn't provide them with all the nutritional requirements they need. I like to feed them Formula 1 flake food as it is a combination of several different nutritional sources and is easy and clean to use (vs. frozen slimey fish parts - which a nice, just messy). If you feed elsewhere in the tank the clown should stay away long enough for the BTA to close on and swallow the food.

Personally I would not try to encourage the anemone to split. If you leave it alone it probably will at some point anyway. Otherwise you just risk injury to the anemone.