I navigated your Websites, and I notice that you already use Toolset successfully:
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The entire site is built with Toolset Layouts and other Toolset components.
Hence I assume you are already quite familiar with this Plugin.
It seems you use a WordPress Multisite Install, if you speak of "subdomains"?
Or are they not built with WordPress as CMS, like the site shared above?
In any case, Toolset cannot be used on a Multisite in terms of sharing it's contents.
You need to:
1. activate Toolset either network wide, or on each sub install,
2. each install will need it's Views, Types, Layouts etc.
You could create them once and use Module Manager to export from one site to another, so you do not need to recreate the same designs.
But you can not use Toolset's contents in a network on all sites, if they are made in one site only.
That said, this is what is possible with the GUI, and what is suggested by us.
Eventually you could achieve your gaol with custom code, but WordPress stores each sub-site's data in a separate place.
Only Users for example, are shared.
Posts are not, and that is where most of Toolset happens, in the posts and meta tables.
This means, that even if you call a certain layout or a View, or simply just a post, with a Custom Database Query so to apply it somewhere, this is:
1. expensive in terms of performance,
2. it's not something we could provide Support for.
You would need to write your own queries and grab the right data from the right database.
You would eventually spare some time in the HTML design, since that could be done with the Layout Plugin, but you still need to call the contents manually, unless you design each sub site in the WP Admin with the Plugins.
As said, we offer the Module manager, this is very helpful to design things once, and then apply them to several sites by importing the created Toolset contest and designs.
About integrating Layouts, in any WordPress Theme that uses the_content() Toolset Layouts will replace exactly that part of the Code.
But, if you are willing to integrate it with manual PHP (it's not a complex code), then you can also style the rest of your template, by just adding the_ddlayout() to the Template you want to design with Toolset Layouts.
This function supports as well a Layout Slug (Slug of the Layout), so you can call specific Layouts if you want.
This is usually done in a WordPress theme, since WordPress needs a theme to run.
I can offer you this simple educational Theme which is integrated with Toolset Layouts, you can use it however you like, it is a great Theme to understand WordPress Theme coding in general and Toolset's Integration in particular:
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It will show exactly how you can integrate or call Toolset contents in your PHP code.
If you require specific help on how to replicate the design of the two subsites you shared, with Toolset, I can help with this in a ticket for each issue.
Please let me know if I missed anything, and do not hesitate to open tickets for single issue you eventually stumble upon using Toolset.