I suppose you should be able to divide it up as a potato as you say--taking some eyes with each piece (though I've never done it this way--usually easily and cheaply available by me). No, it's not too late--they are FAST growers when happy. Just provide heat and LOTS and LOTS of water--they can even grow in water.

I plant the whole tuber. I wouldn't cut it up. They are cheap enough at the ethnic markets.

my family has grown taro since i was little. just plant it in the ground, about 4/5. leave the very top above ground level. it'll grow like an elephant ear plant.

keep them moist but not sitting in water. they will grow a lot of leaves. once the leaves get huge, about 2 feet long, add extra soil to the base of the plant.

in the fall dig it up and you should find a large tuber and also a couple small ones around it. save the smaller ones and you can plant them next spring.