Your problem here is that you're actually using two regexes. Interpolating these separately works fine:

ken@ganymede: ~/tmp
$ cat pm_re2_file.txt
Hello World

--Begin--
Cat
and
Dog
--End--

other stuff 
This is a test
Good Bye
ken@ganymede: ~/tmp
$ RE1="--Begin--" RE2="--End--" perl -00 -ne 'print if /$ENV{RE1}/ && 
+/$ENV{RE2}/' pm_re2_file.txt
--Begin--
Cat
and
Dog
--End--

ken@ganymede: ~/tmp
$
[download]

If you have a specific requirement to use just one environment variable, I'd suggest using eval:

ken@ganymede: ~/tmp
$ RE1="/--Begin--/ && /--End--/" perl -00 -ne 'print if eval $ENV{RE1}
+' pm_re2_file.txt
--Begin--
Cat
and
Dog
--End--

ken@ganymede: ~/tmp
$
[download]