Do I have to worry about temperature with neodymium magnets?

Yes.  Neodymium Iron Boron magnets are sensitive to heat.  If a magnet heated above its maximum operating temperature (176°F (80°C) for standard N grades) the magnet will permanently lose a fraction of its magnetic strength.  If they are heated above their Curie temperature (590°F (310°C) for standard N grades), they will lose all of their magnetic properties.  Different grades of neodymium different maximum operating and Curie temperatures.

What does “Magnetized thru thickness” mean?

We use the description "Magnetized thru thickness" to identify the locations of the poles on our block magnets.  The thickness is always the last dimension listed for block magnets.  If you take one of our block magnets and place it on a flat surface with the thickness dimension as the vertical dimension, the poles will be on the top and bottom of the magnet as it sits.  For example:  Our BX082 blocks are 1" x 1/2" x 1/8" thick.  If you place one of the blocks so it is on a flat surface with 1/8" as the vertical dimension, the poles will be on the top and bottom as the magnet sits.  This means the poles are located in the middle of the 1" x 1/2" sides.