Hi, Ray. You got some good questions and, unfortunately, the answers will not be what you hoped. Lets start with the bugs. The only way to kill them is with heat. You have to put them in an environment where the internal temperature can get up to 135 degrees F. for a day or so. Chemical killers will stay on the outside and not get to where the bugs live.
The heat can come from a kiln and, once the wood is below 20% moisture, any future bugs will not like it. In this area, the chances of termites are slim, but powder post beetles, carpenter ants and a bunch of others are possible.
With a section of trunk like you have, the probability of splitting is almost certain as the wood dries. The moisture from the sap that was flowing from the ground to the leaves to keep the tree alive and healthy was a large amount, probably about 40% by weight, of the mass of the tree. From the time the tree was cut and the cells were in a "less wet" environment, the cells have been shrinking and they will continue to do so from the outside of the piece toward the inside. Since the outside is shrinking but the inside is not, the outside becomes stressed and eventually cracks. You see this with a pile of firewood where almost every piece splits eventually and this helps you to slpit the wood totally for the fireplace.
You can dry thinner sections by putting them in a drier environment, like stacked with sticks between the slices, outside but under cover, and get most of the water and insect activity out. Figure on one year per inch of thickness.
You can also control the splitting by sawing a split yourself where you want it to be. Something else that you can do is to saw the slices up into pie-slice like sections now and glue them together when they are dry. Keep the pie slices in order and figure on losing a pie slice or more to complete a full circle later by combining horizontal slices.
As far as the tree with the rotten center, I would not mess with that. You do not know how far the rot extends and the missing wood isn't as pretty as real wood.
You can get rid of the moss with a mixture of 1 part household bleach to 5 parts water and a scrub brush. This will be dangerous to your eyes, so wear good eye protection. You may also plan on never wearing the clothes you are wearing for this again unless you like the little white spots that will be all over them as the bleach works on your clothes too. Be careful. Keep a flowing water hose handy and rinse you and the tree bark when you have removed the moss. Read the instruction on the bleach bottle and comply with the warnings. The bark may not stay on in any case as the slice drys and moves.