Name change and Hall of Monuments

i know this may seem like a stupid question but if i change my characters name does the stuff in the hall of monuments reflect my characters new name or my old one such as when i enter the hall will it say (character original name) or (character new name).

It changes with your namechange. Ive done the same.

Weapons still stay whatever you customized them as though...

In which case .. can you still use them (with a name that's no longer yours) or do you have to get new weapons ?

Same question with armors and books (is there something else that can be customized ?)

I changed my name a few weeks ago after having customized my weapons and ofcourse my armor (duh ) and I can still use everything that was customized for my old name. The HoM changes to reflect your new name and all your weapons will still be able to be used by you. At least, thats been the case for me.

Since items in the Hall of Monument are account based and not based off any specific character, you won't lose anything for deleting a character, or renaming them.

As well, any items that are customized to your character, are customized to an ID, rather than a name. Each character you make has a unique ID associated with them, and that is what the customized items relate to. That is why, if you delete a character that has customized items, and remake the same character with the same name, they can't use them. The items are customized to the ID of the old character.