How to display custom taxonomy tags OUTSIDE the loop

I think the get_terms() function will do what you want:

$suburbs = get_terms('suburbs','hide-empty=0&orderby=id');
shuffle($suburbs);
foreach ( $suburbs as $suburb ) {
   if( ++$count > 10 ) break;
   echo "COUNT:$count ID:$suburb->term_id NAME:$suburb->name SLUG:$suburb->slug <br />";
}

What do you want them to link to? Can you give an example of a link?

As a guess, I think this might be what you want:

$suburbs = get_terms('suburb','hide-empty=0&orderby=id');
shuffle($suburbs);
foreach ( $suburbs as $suburb ) {
   if( ++$count > 10 ) break;
   echo '<li><a href="'.get_term_link($suburb).'">'.$suburb->name.'</a>';
}

I’m not too good at mind reading. What kind of separator? Show me how you want it to look.

Here is one way to do it.

$suburbs = get_terms('suburb','hide-empty=0&orderby=id');
shuffle($suburbs);
foreach ( $suburbs as $suburb ) {
   if( ++$count > 10 ) break;
   echo $sep . '<a href="'.get_term_link($suburb).'">'.$suburb->name.'</a>';
$sep = ' | ';  // Put your separator here.
}

Did you actually try the code I posted? The separator does not appear before the first entry, or after the last entry. Actual output on my test site:

$10,000 - $60,000 | Resort | Fort Wayne | One under Gull | Odd Pages | Gosh another test | bcategory | Cities | Low Price | $100,000 - $149,999

There is no hyphen in the code I supplied. When I copy the code you showed above and run it, I get no hyphen. You must look in the code before what you showed to find it.