Points one and three mostly correct. Medical/biological waste is generally shipped out by disposal companies (in those big red biohazard bags) and incinerated.
Amputated limbs and organs are collected and stored in the pathology department until they can be incinerated.
My uncle runs the Gross Anatomy department at the local Med College. So, from the horses mouth.
Part of your appendix almost certainly still exists.
Hospital pathology departments typically retain the representative sections taken from surgical specimens for at least ten years, under professional guidelines. The tissue is stored embedded in paraffin blocks that were used (and can be reused) to make the thin sections placed on glass slides and stained for examination under a microscope. The slides themselves are kept for extended periods also. So your appendix’s final moments live on, so to speak.
While long-term access to appendiceal tissue is unlikely to be medically relevant, such is not the case for other surgical pathology specimens, most notably tumors. Tests may be developed that have prognostic or treatment significance, so it’s useful to keep tissue blocks in storage for a long time so new tests can be performed.