Answer

Can a coding agent add a custom CRM object without me writing page code?

For CRUD, yes: a module manifest becomes a migration, a service, a REST resource, an SDK method and Admin screens with no page code (C-01), and one field declaration gives a generated object a foreign key, runtime target validation, schema metadata and an Admin selector (C-02). References are generated-to-generated many-to-one only — many-to-many, inverse collections, cascade delete and generated-to-core references do not exist. What is not generated is the interesting half: workflows and approvals for a custom object are still handwritten, and pipeline stages are proven on the built-in Opportunity module, with configurable pipelines for generated custom objects not claimed (C-05). Changing an object later goes through explicit revisions, a checked-in state file and append-only named migrations, but that view is source-only — what a particular database has actually applied is not knowable from it (C-19).