It seems like the Display Suite module completely ignores
hook_process_node(). Is this the case?
If it is so, what is the way to make minor changes to the final rendering with access to all the node variables?
Display Suite does indeed hijack the node rendering to a point where the default preprocess functions may not be used.
The module comes with a set of its own preprocess functions that are documented in ds.api.php.
In this case,
hook_ds_pre_render_alter() can be used instead of the
hook_preprocess_node() to modify the output via the complete render array. Also, the
$entity object is available with all values.