The Weather, Timer and Habits app support widgets. At this time there is no widget for the Calculator app.
Widgets are supported in iOS14 and above.
This is a known bug in iOS14/15.
This is a known bug in iOS14/15. Try removing the widget and re-adding it to your home screen. If that does not work, remove the widget and restart your phone.
The weather widgets update on a fixed frequency, somewhere around twice an hour. Sometimes when you open the app it will get more up to date values than the widget. It should update the widget when it's opened. Sometimes in rare cases the widgets stop updating. If the widget has not updated for more than a day try deleting and re-adding it.