You need to increase a memory limit for PHP – 16MB is definitely not enough for WordPress. You can try to do that by modifying the wp-config.php file (http://codex.wordpress.org/Editing_wp-config.php#Increasing_memory_allocated_to_PHP) or by modifying PHP settings directly (by editing the php.ini file or by using htaccess).
The plugin requires PHP 5 and your server has probably PHP 4 installed. Ask your hosting provider for an upgrade.
Personally I don’t think server settings could cause this kind of issue.
Post types can have the same display name, but different “ID” (the first param of the register_post_type() function) – maybe that’s the reason?
Are you sure you have activated your theme before importing the demo content?
1. WordPress can’t import posts of a non-existing type. In my opinion it’s a good behavior, because otherwise posts would be imported as regular posts (imagine that mess).
3. No, posts and pages would be imported, only the images import would fail.
I’m the author of Natural theme. First of all, please read the theme documentation – there you can find a lot of useful informations. Second, take a look at sample pages – you can find them in the ‘sample content/pages’ folder. If you still have problems with the theme configuration please send me an e-mail through our profile page.