Attach Galleries to Content Types
There are plenty of approaches to this, the main difference being whether you use the Media module or Drupal 7 core's built in image field. The image field route I think will be simpler as you have to create your own view mode to have custom image sizes with display suite and the 7.x-2.x version is still under heavy development. See the related Media modules for options. There are modules for multiple uploads and creating image galleries.

If you choose to use D7 core's image field you can add galleries to content types with the Views Galleriffic module.

Here are Step by step instructions for D7.

You can also switch between styles.

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For multiple uploads you can try Multiupload Imagefield Widget.

For a remote URL you can try Filefield Sources.

For an "Add image" button you can just create a link to the "/node/add/gallery" node or create a custom form.