Added the public lobby to android. (#125)

This is adapted from kleidis old PR to Azahar. Changes from it:
- Fixed inconsistent button styling in the dialog for connection
- Allowed to hide both empty and full rooms.
- Proper serving of preferred games
- Enables web service for android by default
- Better implementation of multiplayer.cpp that works with oop

Also fixes the room network class and turns it into a static namespace
in network

Signed-off-by: Aleksandr Popovich <alekpopo@pm.me>

Co-authored-by: swurl <swurl@swurl.xyz>
Reviewed-on: https://git.eden-emu.dev/eden-emu/eden/pulls/125
Co-authored-by: Aleksandr Popovich <alekpopo@pm.me>
Co-committed-by: Aleksandr Popovich <alekpopo@pm.me>

src/yuzu/multiplayer/client_room.h

@@ -1,6 +1,9 @@
 // SPDX-FileCopyrightText: Copyright 2017 Citra Emulator Project
 // SPDX-License-Identifier: GPL-2.0-or-later
 
+// SPDX-FileCopyrightText: Copyright 2025 Eden Emulator Project
+// SPDX-License-Identifier: GPL-3.0-or-later
+
 #pragma once
 
 #include "yuzu/multiplayer/chat_room.h"
@@ -13,7 +16,7 @@ class ClientRoomWindow : public QDialog {
     Q_OBJECT
 
 public:
-    explicit ClientRoomWindow(QWidget* parent, Network::RoomNetwork& room_network_);
+    explicit ClientRoomWindow(QWidget* parent);
     ~ClientRoomWindow();
 
     void RetranslateUi();
@@ -35,5 +38,4 @@ private:
 
     QStandardItemModel* player_list;
     std::unique_ptr<Ui::ClientRoom> ui;
-    Network::RoomNetwork& room_network;
 };