Hi all,
Ideally you need to remove the bark, there are couple of reasons for this. First is bark is really light (cork is a bark) and secondly if the bark is still fairly firmly attached the wood is still "green", and will contain sugars etc. If the bark isn't firmly attached it will fall of quickly anyway. Some detached bark is useful and long lasting, I've used Oak and Robinia (tie it to a slate if you want it to sink), and they last several years.
With your wood I would put it into soak in water (nothing added)for a couple of weeks, then give it a good scrub with a wire bristled brush (you can buy these cheaply from places that sell motor accessories, they are the ones they use for cleaning up the engine exteriors stc.), after all the loose bits have come off, go over it with a flat bladed screw-driver trying to find any soft bits. Scrape of the soft bits and its ready to use.
Out of your 3 bits, the root and lower trunk look good, the bit with the white fungi looks like it might be a bit soft, I'd also remove the Ivy, I've never used it, but it may well still contain toxins. The root looks like it isn't lignified, so probably won't be any good.
cheers Darrel