What do you do with the wither rose?

The wither rose is dropped when a mob is killed by the wither. Besides being able to be crafted into black die and potentially being used for a semi-automatic mob farm, I can’t think of any uses. However, there is also potential for an iron farm involving iron golems. (I will note that this is extremely impractical as wither roses are a very rare drop and there are much easier ways to make above items.)

Wither Roses can be made into black dye (Which is ludicrously impractical in the early game unless you are in an old style world with zero potential squid spawning zones, in which case you may be better off with a new world) and used for semi-automatic mob farms, as mentioned by Zyxilef. But, they can also be used for traps, and can be used to make suspicious stew that gives the Wither effect. Also, to answer your second question, you cannot use it to make a Wither Skeleton farm. Wither Skeletons are immune to the Wither effect. They can also just be a cool decoration, a trophy item, or a flex that you're rich enough to have them.