Should you spray or paint anything on your tree slices to preserve them or to help stop the bits and pieces that fall off?
Nature will take it's course! They do make water proofing sealers for wood. Check your local hardware store.
Are they to be used inside or out? I just cut some Juniper into coasters and trivets. Coasters are pretty dry, the trivets will need to dry out before using any kind of sealer. Sealing before dry will prevent the wood from drying. Putting little feet on the trivets, with cork on the bottom of feet, also using cork for the bottom of the coasters.
Juniper trivets, will nail gun the little feet to the bottom