It looks like you are reading in the questions file one line at a time rather than parsing them by the === separator. I would do something more like this:

my @lines = <QUESTIONS>; # The entire file in an array
my @questions = (); # A new array for questions
my $tmp = ""; # A tmp variable to capture stuff in
my $count = 0;
foreach my $line(@lines) {
     if($line =~ /^===$/) {
          push(@questions,$tmp) unless $count == 0;
          $tmp = "";
          next;
     }
     $tmp .= $line;
}
# and then to get the last one
push(@questions,$tmp) if $tmp;
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Then to get your random stuff

my @rand_questions = ();
foreach (0 .. $num_questions_per_test) {
     my $rand = int(rand(scalar(@questions) - 1);
     push(@rand_questions,$questions[$rand]);
}
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That would put all the random questions in the array rand_questions. I'm sure there are ways to make it simpler I tend to code the long way so I can understand it when I look at it 6 months later :) hope that helps.