chore: add TypeScript support

Added TypeScript configuration for Vite and Node (tsconfig.app.json and
tsconfig.node.json, referenced in tsconfig.json).
Vite configuration was migrated to TypeScript.
Added ESLint TypeScript rules and general ESLint TypeScript
configuration.
Npm scripts for type checking have been added and build script has been
updated to run type checking script before building.

tsconfig.app.json

@@ -0,0 +1,32 @@
+{
+    "compilerOptions": {
+        "target": "ES2020",
+        "lib": ["ES2020", "DOM", "DOM.Iterable"],
+        "module": "esnext",
+        "moduleResolution": "bundler",
+
+        "strict": true,
+        "noEmit": true,
+        "isolatedModules": true,
+        "verbatimModuleSyntax": true,
+        "noImplicitAny": true,
+
+        "allowJs": true,
+        "checkJs": false,
+
+        "resolveJsonModule": true,
+        "skipLibCheck": true,
+        "types": ["vite/client"],
+
+        "paths": {
+            "@/*": ["./src/*"]
+        },
+
+        "tsBuildInfoFile": "./node_modules/.tmp/tsconfig.app.tsbuildinfo",
+
+        "useDefineForClassFields": true,
+        "jsx": "preserve",
+        "esModuleInterop": true
+    },
+    "include": ["src/**/*.ts", "src/**/*.js", "src/**/*.vue"]
+}