Does the STDERR get closed automatically when I fork?

It's really trivial to test:

$ perl -wle 'fork; warn $$'
1787 at -e line 1.
1786 at -e line 1.
$
[download]

So, the answer is no.

Also, from perldoc -f fork

File descriptors (and sometimes locks on those descriptors) are shared, while everything else is copied.