docs: fix cargo doc, add to CI check, and pr recipe

[storopoli]

Description

• chore: add rustdocs in the pr recipe
• doc: fix cargo doc lints
• ci: add docs check in PRs and merge groups
• chore: don't run cargo doc in dependencies

Type of Change

• Bug fix (non-breaking change which fixes an issue)
• New feature/Enhancement (non-breaking change which adds functionality or enhances an existing one)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• Documentation update
• Refactor

Checklist

• I have performed a self-review of my code.
• I have commented my code where necessary.
• I have updated the documentation if needed.
• My changes do not introduce new warnings.
• I have added tests that prove my changes are effective or that my feature works.
• New and existing tests pass with my changes.

Related Issues

Closes STR-337

[Rajil1213]

Looks good. Maybe remove the compress and archive steps from the workflow file while you're at it.

Otherwise, you can merge this once the CI checks are done (might fail due to the functional tests).

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 64.53%. Comparing base (7aab4b7) to head (7289e46).
Report is 6 commits behind head on master.

@@            Coverage Diff             @@
##           master     #297      +/-   ##
==========================================
- Coverage   64.64%   64.53%   -0.11%     
==========================================
  Files         187      187              
  Lines       19712    19712              
==========================================
- Hits        12742    12722      -20     
- Misses       6970     6990      +20     

Files with missing lines	Coverage Δ
crates/bridge-sig-manager/src/errors.rs	100.00% <ø> (ø)
crates/bridge-sig-manager/src/manager.rs	98.41% <ø> (ø)
crates/bridge-tx-builder/src/context.rs	100.00% <ø> (ø)
crates/bridge-tx-builder/src/deposit.rs	97.44% <ø> (ø)
crates/bridge-tx-builder/src/errors.rs	12.50% <ø> (ø)
crates/bridge-tx-builder/src/operations.rs	96.51% <ø> (ø)
crates/btcio/src/inscription.rs	93.96% <ø> (ø)
crates/btcio/src/rpc/error.rs	0.00% <ø> (ø)
crates/consensus-logic/src/state_tracker.rs	46.40% <ø> (ø)
crates/db/src/entities/bridge_tx_state.rs	96.61% <ø> (ø)
... and 13 more

... and 1 file with indirect coverage changes

[storopoli]

Hey I've added 0.01% in the code coverage without touching code :)
The magic of good documentation 🪄

[Rajil1213]

Approving these changes as the functional tests were already broken. But we might want to merge this once those are fixed and this branch has been rebased.