Close remaining quality-gate gaps: web/server + root-file format, cli/tui __tests__ typecheck #1767

Follow-up to #1689 (deferred from PR #1766 by agreement in review). The same class of gap that #1689 closed still exists in a few spots:

• clients/web/server/** is linted but not prettier-checked. web's format:check is prettier --check src, so the server/ dir (Node backend) is eslinted (eslint .) but never format-gated.
• Root-level files are neither formatted nor linted: scripts/**, test-servers/src/**, vitest.shared.mts, and the new root eslint.config.js.
• clients/{cli,tui}/__tests__ are not typechecked. The new typecheck scripts (Close quality-gate coverage gaps: typecheck cli/tui, format+lint core/, coverage core/json+core/client #1689) include src only (exclude: ["**/*.test.ts"]), so the test dirs — which do the most as casting — get no tsc pass.

Proposed

• Extend web format/format:check to include server.
• Add a root format/lint pass (or fold into an existing scope) covering scripts/, test-servers/src, vitest.shared.mts, eslint.config.js.
• Add a second tsconfig (or widen include) so cli/tui __tests__ are typechecked, and fix anything it surfaces.

None of this is behavior-changing; it's gate coverage, same as #1689.