Why are these queries slow in some environments and not others? Can we fix their performance?
You may want to disable the variable innodb_stats_on_metadata.

According to the MySQL Documentation on innodb_stats_on_metadata


  When this variable is enabled (which is the default, as before the
  variable was created), InnoDB updates statistics during metadata
  statements such as SHOW TABLE STATUS or SHOW INDEX, or when accessing
  the INFORMATION_SCHEMA tables TABLES or STATISTICS. (These updates are
  similar to what happens for ANALYZE TABLE.) When disabled, InnoDB does
  not update statistics during these operations. Disabling this variable
  can improve access speed for schemas that have a large number of
  tables or indexes. It can also improve the stability of execution
  plans for queries that involve InnoDB tables.