refactor: improve router type safety and usage

- Add explicit Router type imports across views
- Replace $router type casting with proper typing
- Use $router.back() instead of $router.go(-1) for consistency
- Add proper route and router typings to components
- Clean up router navigation methods
- Fix router push/back method calls

This commit improves type safety and consistency in router usage across
the application's view components.

src/App.vue

@@ -40,7 +40,10 @@
             <div
               class="flex items-center justify-center w-12 bg-slate-600 text-slate-100"
             >
-              <font-awesome icon="circle-info" class="fa-fw fa-xl"></font-awesome>
+              <font-awesome
+                icon="circle-info"
+                class="fa-fw fa-xl"
+              ></font-awesome>
             </div>
 
             <div class="relative w-full pl-4 pr-8 py-2 text-slate-900">
@@ -63,7 +66,10 @@
             <div
               class="flex items-center justify-center w-12 bg-emerald-600 text-emerald-100"
             >
-              <font-awesome icon="circle-info" class="fa-fw fa-xl"></font-awesome>
+              <font-awesome
+                icon="circle-info"
+                class="fa-fw fa-xl"
+              ></font-awesome>
             </div>
 
             <div class="relative w-full pl-4 pr-8 py-2 text-emerald-900">
@@ -86,7 +92,10 @@
             <div
               class="flex items-center justify-center w-12 bg-amber-600 text-amber-100"
             >
-              <font-awesome icon="triangle-exclamation" class="fa-fw fa-xl"></font-awesome>
+              <font-awesome
+                icon="triangle-exclamation"
+                class="fa-fw fa-xl"
+              ></font-awesome>
             </div>
 
             <div class="relative w-full pl-4 pr-8 py-2 text-amber-900">
@@ -109,7 +118,10 @@
             <div
               class="flex items-center justify-center w-12 bg-rose-600 text-rose-100"
             >
-              <font-awesome icon="triangle-exclamation" class="fa-fw fa-xl"></font-awesome>
+              <font-awesome
+                icon="triangle-exclamation"
+                class="fa-fw fa-xl"
+              ></font-awesome>
             </div>
 
             <div class="relative w-full pl-4 pr-8 py-2 text-rose-900">