fix: add http server timeouts to prevent fd exhaustion

[boidushya]

Overview

The server started with http.ListenAndServe, which has no connection timeouts, so idle and half-dead keep-alive connections were never closed. They leaked file descriptors over weeks until the process hit its 65k limit, stopped accepting connections, and took the whole API down. Gave the server explicit read, write, header, and idle timeouts so every connection has a bounded lifetime, and exempted the cache-db dump endpoint since it legitimately streams for longer. Added a file-descriptor monitor too, so the next slow climb pages through the notification channels at 80% of the limit instead of going dark at 100%.

[better-lyrics-harmonizer]

Preview torn down

This PR's ephemeral environment was cleaned up.

Detail	Value
Status	Torn down
Branch	fix/server-timeouts-fd-leak
Last commit	a0db8fb
Closed	2026-07-03 13:16:12 UTC
Workflow	run #28663028116

_{Reopen the PR if you need to spin it back up.}