Using Context, can I (programiatically) move the page's menu to a different region?
You could place all blocks in template_preprocess_page with 

$variables['page'][$region_key] = array();
$variables['page'][$region_key][]['#markup'] = $region_content;


Seeing your edit, you can check for the plugins blocks like this

$context_content_bottom = array();
  $blocks_content_bottom = array();

  // Blocks that are assigned to the region using Context
  if ($plugin = context_get_plugin('reaction', 'block')) {
    $context_content_bottom = $plugin-&gt;block_get_blocks_by_region('content_bottom');
  }

  // Blocks that are assigned to the region using the blocks interface
  if ($blocks = block_get_blocks_by_region('content_bottom')) {
    $blocks_content_bottom = $blocks;
  }

  // Merge them together
  $vars['region']['content_bottom'] = array_merge($context_content_bottom, $blocks_content_bottom);


Taken from here

but why not do it in the UI ??
just place the block several times under opposed conditions