Documentation CI/CD stability and QoL improvements

[NickVolynkin]

Documentation CI/CD stability and quality-of-life improvements.

• Cancel outdated workflow runs in documentation projects #3234
• Use tarantool/actions/prepare-checkout in doc projects with submodules #3237
• Improve deployment webhook reliability and testability in doc projects #3238
• Use fail-fast options in documentation build and deployment scripts
• Report failures in doc projects to a VK Teams chat
• Update all doc projects to the newest version of tarantool/doc-builder
• Deploy latest and 1.10 branches to test servers as well #2443

Template for subtasks:

- [ ] [tarantool/doc@`latest`](https://github.com/tarantool/doc/tree/latest)
- [ ] [tarantool/doc@`tt`](https://github.com/tarantool/doc/tree/tt)
- [ ] https://github.com/tarantool/enterprise_doc/tree/master
- [ ] https://github.com/tarantool/tdg-doc/tree/latest (2.x)
- [ ] https://github.com/tarantool/tdg-doc/tree/master (1.x)
- [ ] https://github.com/tarantool/tarantool-internals/tree/master