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Error on activation: "Parse error: syntax error, unexpected..."

Make sure your host is running PHP 5. The only foolproof way to do this is to add this line to wp-config.php (after the opening <?php tag):

var_dump(PHP_VERSION);

When I go to page 2, I see the same posts as on page 1!

You're using query_posts() wrong. See The Right Way To use query_posts()

Does PageNavi work with secondary WP_Query instances?

Yes; read this tutorial

How do I ignore the options page?

If you are running a multi-language plugin, you will probably want to ignore the strings in the options page.

You can do that like so:

<?php wp_pagenavi( array( 'options' => PageNavi_Core::$options->get_defaults() ) ); ?>