feat: initial project structure and generate API types

Initialized `package.json`, add tsup build config (CJS/ESM/DTS), wire up
full Makefile targets (lint/test/check/build), add missing BadRequest
response component to OpenAPI spec, generate TypeScript types from spec,
configure biome to exclude generated files, and add `@types/ws`

biome.json

@@ -0,0 +1,35 @@
+{
+	"$schema": "https://biomejs.dev/schemas/2.4.14/schema.json",
+	"vcs": {
+		"enabled": true,
+		"clientKind": "git",
+		"useIgnoreFile": true
+	},
+	"files": {
+		"ignoreUnknown": false,
+		"includes": ["**", "!src/models/generated.ts", "!dist"]
+	},
+	"formatter": {
+		"enabled": true,
+		"indentStyle": "tab"
+	},
+	"linter": {
+		"enabled": true,
+		"rules": {
+			"recommended": true
+		}
+	},
+	"javascript": {
+		"formatter": {
+			"quoteStyle": "double"
+		}
+	},
+	"assist": {
+		"enabled": true,
+		"actions": {
+			"source": {
+				"organizeImports": "on"
+			}
+		}
+	}
+}