Well…a number of things here.
There is LockHTML which will let you put any HTML on the lockscreen, which includes all iWidgets - and a number of iWidgets are clock/weather variations. There is Forecast, which is a weather tweak that has been around since iOS 6. It will place the forecast on the lockscreen and pressing on the temperature will take you to the right to a second page that will show the forecast. That requires LockPages to be installed, but you can disable this feature if you wish.
Weatherboard will take the live weather from the Weather app and overlay it on to the lockscreen, homescreen(s) or both.
But I know of nothing specific that would take you direct to the Clock app or direct to the Calendar app, especially if you have a lockscreen code. Activator might work for that, or maybe someone else could suggest something.
Here is what I have…
On my lockscreen, Weatherboard for the live (animated) weather. Forecast for the forecast (it also hides my clock and date, but you can change that, it's an option in Forecast) and LockHTML to have the clock (an iWidget) above Forecast. Also is the second lockscreen for Forecast.
On my homescreen I have iWidgets, Weatherboard and a modified iWidget.
Hope that gives you some ideas/options.
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