What happens if you have the central heating on at the same time as hot water? Does the boiler still short cycle? Does the cylinder reach target temperature?
I reckon you're not losing enough heat into the cylinder for the boiler return temperature to drop sufficiently, hence your boiler reaches a point where it can't modulate down any lower and thus cuts off the burner.
No the regeneration process happens inside the unit so you won't see any evidence that it's happening apart from the noise.
The salt can be filled right up to the top of the chamber - only a small amount of water enters here and it's at the bottom in order to make brine to backwash the resin...
Looks like a Wickes E10T. Good little unit- I have one myself. Use salt tablets in it, just pour up to the top of the lidded compartment every month or so. A 25kg bag of salt tablets costs a tenner or so.
The noise you're hearing is the regeneration cycle which is completely normal and I have...