When your iPhone receives a WhatsApp/Telegram push notification e.g.
wife:
\"buy pizza\"
Question1: Is it that the app has
First, downloading the payload usually isn't a problem. I'd imagine that the payload is usually very small (probably <1Kb for short text messages). In fact, the maximum size for a regular remote payload is 4KB (5KB for VoIP notifications). See Creating the Remote Notification Payload.
Second, it is hard to have your app access any remote notification data. The only way that is possible is through a Silent Notification, which is not recommended for simple text notifications. A Silent Notification wakes up your app and gives it 30 seconds to perform actions in the background before it is shut down again. You could potentially send a remote Silent Notification that causes your app to manually trigger a regular, local notification based on that remote one (or send both a silent and regular remote notification), but again, such use of Silent Notifications is not recommended:
Silent notifications are not meant as a way to keep your app awake in the background, nor are they meant for high priority updates. APNs treats silent notifications as low priority and may throttle their delivery altogether if the total number becomes excessive. The actual limits are dynamic and can change based on conditions, but try not to send more than a few notifications per hour.
See Configuring a Silent Notification for more info.
Another possible way that apps could "download" remote notifications as they are received is described in Modifying the Payload of a Remote Notification. You could simply save the notifications and choose not to modify them.
There is no universal yes/no answer that works for all apps to either of your questions. Instead, there are a number of different ways that apps could receive remote notifications and process their data.
EDIT: You should also implement application(_:didReceiveRemoteNotification:fetchCompletionHandler:) and retrieve/save data from the user info of the notification so that the data is available faster in your app. However, that method is not guaranteed to be called. So you could/should implement it, but more importantly, have your app check for new data from your own server so that nothing is missed.