You can certainly achieve this with WordPress.
Creating an image gallery on a page should be straightforward, organizing pages and creating menu is as well. For logging in you can use the default one (at yoursite.com/wp-admin) and then installing a plugin to forward users to a certain page upon logging in. Another option is creating your own custom login page. There’s lots of information online on that as well; here’s a decent tutorial:
http://www.wpbeginner.com/plugins/how-to-create-custom-login-page-for-wordpress/
In general, I would recommend that you attack one task at a time. That is, first figure out your page structure, then figure out creating an image gallery, then – commenting on images. Login/forwarding to a certain page part is a separate task that you can either do first or last, depending on whether you prioritize more complex tasks.
The learning curve may be steep at times, but it pays off every time. Good luck!
1. Yes.
2. If you gallery directs images to the attachment page, it’s easy because attachments support comments. I’m sure there are plugins to do this in a light box (normally expected gallery format) as well.
3. If you restrict comments to logged in users, the comment link will prompt for a login and/or registration, so you don’t have to code that yourself.
4. One step at a time. Set up a basic WP site with a fixed front page. Then add a gallery to it. There are many, many (many!) plugins that provide galleries.
Some resources
https://premium.wpmudev.org/blog/wordpress-gallery-plugins/
https://wordpress.org/plugins/search.php?type=term&q=gallery+comments
