Refactor: Extract Plex composables and smaller components

Split large PlexSettings component into reusable pieces:

Composables:
- usePlexApi.ts: API functions for user data, servers, libraries
- usePlexAuth.ts: OAuth authentication flow, PIN generation, polling

Components:
- PlexAuthButton.vue: Sign-in button with OAuth popup
- PlexProfileCard.vue: User profile with badges (Pass, 2FA, Labs, years)

Benefits:
- Better code organization and maintainability
- Reusable authentication logic
- Cleaner separation of concerns
- Easier testing and debugging

src/components/plex/PlexAuthButton.vue

@@ -0,0 +1,149 @@
+<template>
+  <div class="plex-connect">
+    <div class="info-box">
+      <IconInfo class="info-icon" />
+      <p>
+        Sign in to your Plex account to get information about recently added
+        movies and to see your watch history
+      </p>
+    </div>
+    <div class="signin-container">
+      <button @click="handleAuth" :disabled="loading" class="plex-signin-btn">
+        <svg
+          v-if="!loading"
+          width="20"
+          height="20"
+          viewBox="0 0 256 256"
+          fill="currentColor"
+        >
+          <path
+            d="M128 0C57.3 0 0 57.3 0 128s57.3 128 128 128 128-57.3 128-128S198.7 0 128 0zm57.7 128.7l-48 48c-.4.4-.9.7-1.4.9-.5.2-1.1.4-1.6.4s-1.1-.1-1.6-.4c-.5-.2-1-.5-1.4-.9l-48-48c-1.6-1.6-1.6-4.1 0-5.7 1.6-1.6 4.1-1.6 5.7 0l41.1 41.1V80c0-2.2 1.8-4 4-4s4 1.8 4 4v84.1l41.1-41.1c1.6-1.6 4.1-1.6 5.7 0 .8.8 1.2 1.8 1.2 2.8s-.4 2.1-1.2 2.9z"
+          />
+        </svg>
+        {{ loading ? "Connecting..." : "Sign in with Plex" }}
+      </button>
+      <p class="popup-note">A popup window will open for authentication</p>
+    </div>
+  </div>
+</template>
+
+<script setup lang="ts">
+  import { usePlexAuth } from "@/composables/usePlexAuth";
+  import IconInfo from "@/icons/IconInfo.vue";
+
+  const emit = defineEmits<{
+    authSuccess: [token: string];
+    authError: [message: string];
+  }>();
+
+  const { loading, openAuthPopup } = usePlexAuth();
+
+  function handleAuth() {
+    openAuthPopup(
+      token => emit("authSuccess", token),
+      error => emit("authError", error)
+    );
+  }
+</script>
+
+<style lang="scss" scoped>
+  @import "scss/variables";
+  @import "scss/media-queries";
+
+  .info-box {
+    display: flex;
+    gap: 0.65rem;
+    padding: 0.65rem;
+    background-color: var(--background-ui);
+    border-radius: 0.25rem;
+    margin-bottom: 0.85rem;
+    border-left: 3px solid var(--highlight-color);
+
+    @include mobile-only {
+      padding: 0.6rem;
+      gap: 0.55rem;
+      margin-bottom: 0.7rem;
+    }
+
+    p {
+      margin: 0;
+      font-size: 0.9rem;
+      line-height: 1.4;
+
+      @include mobile-only {
+        font-size: 0.85rem;
+      }
+    }
+  }
+
+  .info-icon {
+    width: 20px;
+    height: 20px;
+    fill: var(--highlight-color);
+    flex-shrink: 0;
+
+    @include mobile-only {
+      width: 18px;
+      height: 18px;
+    }
+  }
+
+  .signin-container {
+    display: flex;
+    flex-direction: column;
+    gap: 0.6rem;
+  }
+
+  .plex-signin-btn {
+    padding: 1rem 1.75rem;
+    background-color: #c87818;
+    color: $white;
+    border: none;
+    border-radius: 0.75rem;
+    font-size: 1.1rem;
+    font-weight: 700;
+    cursor: pointer;
+    transition: all 0.2s;
+    display: inline-flex;
+    align-items: center;
+    justify-content: center;
+    gap: 0.75rem;
+    align-self: flex-start;
+    box-shadow: 0 4px 12px rgba(200, 120, 24, 0.25);
+    letter-spacing: -0.01em;
+
+    @include mobile-only {
+      width: 100%;
+      padding: 0.9rem 1.4rem;
+      font-size: 1rem;
+    }
+
+    &:hover:not(:disabled) {
+      background-color: #b36a15;
+      transform: translateY(-2px);
+      box-shadow: 0 8px 20px rgba(200, 120, 24, 0.4);
+    }
+
+    &:active:not(:disabled) {
+      transform: translateY(0);
+      box-shadow: 0 4px 12px rgba(200, 120, 24, 0.3);
+    }
+
+    &:disabled {
+      opacity: 0.6;
+      cursor: not-allowed;
+    }
+
+    svg {
+      flex-shrink: 0;
+      width: 22px;
+      height: 22px;
+    }
+  }
+
+  .popup-note {
+    margin: 0;
+    font-size: 0.85rem;
+    opacity: 0.65;
+  }
+</style>