You can't. The actual image will be saved to a rtaImage, and the file name will be lost as a result. You can still determine the type by checking the headers for the image when you retrieve the URL (Content-Type), but that's it.
The Tooling API uses a different base URI to the standard REST API: Docs
The base URI for each Tooling API REST resource is http://domain/services/data/vXX.X/tooling/ where domain is a My Domain or a custom domain and vXX.X is the API version number. For example: https://MyDomainName.my.salesforce.com/services/data/v35.0/tooling/
So, as kurunve commented, ...
Here is what appears to happen. Based on a couple of hours of work so certainly not definitive.
When a new "Digital Experience" is created starting from a template (that includes the theme layout), a copy of the template is created. That works fine for the parts that the page builder makes editable. But the template header is not editable, though ...
This worked fine for me:
I opened up the Rest Explorer in workbench (logged in to dev org as admin) and executed a GET request using the following URI
This returned picklist values:
Perhaps you cannot see the fields because of field level security for the fields? Ensure the user you are ...
Another way to do this programmatically is to use the NPM package sfdc-soup (https://github.com/pgonzaleznetwork/sfdc-soup) which enhances the capabilities (and fixes many of the bugs) of the dependency API
You could use HappySoup.io (a 100% free and community-led open source project) , which enhances the capabilities of the MetadataComponentDependency API and provides impact and dependency analysis for standard fields.