To be clear all of those errors seem to be due to the WordPress Importer plugin itself.
You can tell because all the PHP Notices mention:
/home/user/public_html/wordpress/wp-content/plugins/wordpress-importer/wordpress-importer.php
This is not your fault nor something you will be able to resolve without contributing to the plugin itself (see the final part of this answer if you are interested.)
As of this post the WordPress Importer plugin (Version 0.6.3) is in dire need of maintenance and repair. Take a look at the plugin’s support page and you will see a lot of other people experiencing these same issues.
In many cases the plugin may work but you are likely to experience issues and see PHP errors. I would take it with a grain of salt, meaning, backup before use (as always) and double check everything after you use it.
There is hope however!!!
In 2015 Ryan McCue a co-lead on the REST API project announced he was beginning a rewrite of the plugin dubbed WordPress Importer Redux. You can follow that progress (and contribute!) over at the project’s GitHub repo. Development seems active so I’d keep an eye on that one.
Bonus:
To really test your theme you can look into PHP_CodeSniffer and the PHP_CodeSniffer rules (sniffs) to enforce WordPress coding conventions.