feat(ci): build and check rustdocs for warnings; fix(docs): add missing cfg flag to allow displaying features on doc.rs, fix various warnings

[jlizen]

📬 Issue #, if available:
n/a

✍️ Description of changes:
Fix for: https://docs.rs/crate/lambda_runtime/0.14.0/builds/2162130

[INFO] [stderr]  Documenting lambda_runtime v0.14.0 (/opt/rustwide/workdir)
[INFO] [stderr] error[E0658]: `#[doc(cfg)]` is experimental
[INFO] [stderr]   --> src/lib.rs:35:20
[INFO] [stderr]    |
[INFO] [stderr] 35 | #[cfg_attr(docsrs, doc(cfg(feature = "tracing")))]
[INFO] [stderr]    |                    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
[INFO] [stderr]    |
[INFO] [stderr]    = note: see issue #43781 <https://github.com/rust-lang/rust/issues/43781> for more information
[INFO] [stderr]    = help: add `#![feature(doc_cfg)]` to the crate attributes to enable
[INFO] [stderr]    = note: this compiler was built on 2025-05-26; consider upgrading it if it is out of date
[INFO] [stderr] 
[INFO] [stderr] For more information about this error, try `rustc --explain E0658`.
[INFO] [stderr] error: could not document `lambda_runtime`

We want to display the feature gates on items in docs.rs. However, we were missing the crate-level #![cfg_attr(docsrs, feature(doc_cfg))] necessary to do so.

This made it through our CI since we don't build/validate docs. Adding that in.

I structured it as a standalone step since docs.rs target only builds on the nightly branch. I just tossed all the crates into it rather than segmenting - there are cross-crate doc linking interdependencies, and anyway it's a quick step.

This required fixing a TON of docs warnings since I have the step configured to fail on warnings (broken doc links, etc).

Testing

It's passing on my branch's CI: https://github.com/jlizen/aws-lambda-rust-runtime/actions/runs/15281293634/job/42980737800

Also the regular builds are still passing.

🔏 By submitting this pull request

• I confirm that I've ran cargo +nightly fmt.
• I confirm that I've ran cargo clippy --fix.
• I confirm that I've made a best effort attempt to update all relevant documentation.
• I confirm that my contribution is made under the terms of the Apache 2.0 license.

[jlizen]

Dwarnings results in failing on warnings

[bnusunny]

LGTM