Is it important to determine the type of solder used on a board, and if so, how?
Anything using lead-free solder would (or should) have a ROHS compliancy symbol. Otherwise why would they bother going with lead-free. It's not particularly easy to work with compared to old industry standard leaded solder.

Melting or remelting points vary depending on solder compositions, and there plenty of mixtures that are lead-free that have a lower melting point than leaded.

Practical solution, treat everything you are desoldering as if it has lead-free if its been produced in the last say 10 or 13 years, and don't use your regular soldering iron for desoldering.