To the first problem: Use backreferences with explicit repetition delimiters for ". My solution is as follows:

$rep_contact_info = '<table width ="12">';
$rep_contact_info =~ s/(<\s*table.*?width\s*=\s*("{0,1})).*?(\2.*?>)/$
+{1}300${3}/si;
print $rep_contact_info;
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The quotation mark (if any) is captured in $2 and backreferenced by \2 in the RegEx. The curly braces {0,1} take care that the quotation mark doesn't match too often
To your second problem: Use the braces after the dollar sign, instead before it.