I never remove the nest - they are very precious about it - unless it's pee'd on then you have to remove it but always try and leave a bit of the old dry nest behind, even if it's slightly whiffy. They like to forage for their nesting material too, so best to put it out in a big pile in the cage, but not actually inside the house - they take it there themselves to build the nest and sometimes pouch it to do so. If you put it in the house they haven't been able to get things how they want it (plus it's seen as a bit of an intrusion to the house possibly!)
Likewise with the hoards. If the hoard is removed they will be frantic (and trying to escape is a sign of frantic!) They only feel secure when they have food hoarded and usually eat from their hoards rather than from the food bowl. Most food from the bowl is pouched and taken to the hoard (often under their nest) so they sit in bed and snack
Again I leave the hoard alone for really quite a long time unless it's pee'd in and again if you have to remove it because it's pee'd on then try and leave some dry hoard behind and always replace what you've removed with new food in exactly the same place. To have a happy secure hamster!