Fix "Next" never sending admin logs for rounds outside the cache, show a round's total logs on the UI (#16531)

* Fix next never sending logs for rounds outside the cache

* Show round's total log count on the ui

* Disable next button when waiting for a next response

* Cleanup AdminLogsEui.CurrentRoundId

* Fix popout window width

Content.Server/Database/ServerDbManager.cs

@@ -205,6 +205,7 @@ namespace Content.Server.Database
         IAsyncEnumerable<string> GetAdminLogMessages(LogFilter? filter = null);
         IAsyncEnumerable<SharedAdminLog> GetAdminLogs(LogFilter? filter = null);
         IAsyncEnumerable<JsonDocument> GetAdminLogsJson(LogFilter? filter = null);
+        Task<int> CountAdminLogs(int round);
 
         #endregion
 
@@ -588,6 +589,12 @@ namespace Content.Server.Database
             return _db.GetAdminLogsJson(filter);
         }
 
+        public Task<int> CountAdminLogs(int round)
+        {
+            DbReadOpsMetric.Inc();
+            return _db.CountAdminLogs(round);
+        }
+
         public Task<bool> GetWhitelistStatusAsync(NetUserId player)
         {
             DbReadOpsMetric.Inc();