How do I get a field label by the field name?
The label is set per instance (not per field) so you would use the field_info_instance() function with some appropriate arguments:

$entity_type = 'node';
$bundle_name = 'article';
$field_name = 'field_some_field';

$info = field_info_instance($entity_type, $field_name, $bundle_name);

$label = $info['label'];


There's a bunch of other useful functions in the Field Info API documentation that might be helpful in case you haven't seen it already.