The weaver page with post can display custom post type, just enter the slug of the custom Post type in the custom post type filter box of the page with posts options.
To mix different kind of posts on the same page, you can use the Weaver Show Post plugin that will let you define filters to select any type of posts and how to display them, and then give you a shortcode to insert that in any content area.
This allows you to insert and mix several Show post shortcodes with any other content.