You can check for taint checking using $^T (documented in perlvar).

There's no such thing as strict being on or off. There are three kinds of strict checks which are independant. You can check their status by causing a violation in an eval:

my $strict_refs = not eval { no warnings; ${'var'}; 1 };
my $strict_subs = not eval 'no warnings; ThisDoesNotExist; 1';
my $strict_vars = not eval 'no warnings; $ThisDoesNotExist; 1';
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Similar to strict checks, there's no such thing as warnings being on or off. There are many kinds of warning checks which are independant. $^W will tell you if -w on the command line (or equivalent), but each and all individual warnings can be enabled and disabled using warnings without affecting $^W. You can check if a particular warning is enabled using warnings::enabled().