Q: Archive posts to a certain year, then have a generic archive tab for older posts I would like to archive paintings content type by year until 2010, then the rest of the older content types <2010, I would have a generic 'Older Archive' link to list the paintings.
I have already completed the first part:
wp_get_archives( array( 
    'post_type' => 'paintings', 
    'type' => 'yearly', 
    'limit' => 4 ) 
);

It is displaying as:
2017 | 2016 | ...etc
I would like to list those paintings older than 2010 in one header link.

A: As there is no offset parameter in wp_get_archives(), you can simply change the order to do this.
 wp_get_archives( array( 
    'post_type' => 'paintings', 
    'type' => 'yearly', 
    'limit' => 4,
    'order' => 'ASC', ) 
 );

As I understand you case, alphabetical order with year are in descending order (default value). If the first archive year is 2006, change order to ASC will display : 2006, 2007, 2008, 2009. 
If you want to reverse these results, you can play with format and echo parameters and apply your logic to show 2009, 2008, 2007, 2006. But it's another question !
UPDATE (from the comments)
You can use the pre_get_post action to change the query
add_action('pre_get_posts', 'so_41494855');

function so_41494855($query){

   if(is_post_type_archive('paintings') && $query->is_main_query()){
       $before = '2009-12-31';
       $after = '2006-01-01';
       $date_query = array(
           'before' => $before,
           'after' => $after
       );
       $query->set('date_query', $date_query);
       $query->set('posts_per_page', '15');
   }
   return $query;
}

You will find all details for the WP_Query and the date parameter here to adapt you query. 
You will need to detect this archive page with a special archive link. I suggest you to see about add_query_var() and get_query_var() and will be add to the public query variables available to WP_Query.
the conditionnal became :
if(is_post_type_archive('paintings') && get_query_var('show-paintings') == 'older' && $query->is_main_query()){}

Another way to think about this could be clone the custom post type to separate new and old paintings. The conditional to detect the archive page and change the query will became is_post_type_archive('older-painting').
Hope it works & helps !
