Glad updating the Design Manager fixed it.
I’m assuming that designs made with the private setting can not be viewed with your other email address because that address is not the same as the author address, and thus the plugin is trying to only show the design to the author – whether that be in the cart, in the design gallery, on it’s design profile page, etc. If you are logged in as an ADMIN account, though, you should be able to see all designs – private included. Try clearing your cache, clearing your cookies, etc and starting from scratch.
The outbound email that your WordPress site uses is where the MyStyle emails come from, there is not a different setting for this. You should be able to change your email settings in the normal wordpress and woocommerce settings.
If you like, you can try out our Email Manager for even more control over the outbound emails. This add on hooks into the WooCommerce email template for a much better looking email for design-saved confirmations, and allows you to edit the content of that automatic email: