The thermometer is alcohol-based so it presents no hazards other than
the obvious sharp glass fragments. I'll get a replacement out to you the
first of the week.
As to storage, the bottles with the taped caps are chemicals that are
defined as hazardous for shipping purposes. In some cases (like strong
acids) they really do need careful handling, but in many cases they're
defined as hazardous because they're flammable. A tablespoon or two of
rubbing alcohol used to dissolve a stain, for example, is still
technically hazardous, but in reality is no more hazardous than a small
bottle of rubbing alcohol.
You can refrigerate any of the chemicals if you wish, but storing them
in a plastic bin or similar container at room temperature in the dark is
perfectly acceptable.