Tu-4 was indeed a very close copy of B-29, but no, they did not "carefully duplicate the battle damage" on the original. The one prominent example of copying unnecessary things that is usually showcased in this instance is a mistakenly drilled rivet hole in one of the wings that was carefully reproduced thereafter despite there not being any evident purpose for it.
That said, even then Tu-4 wasn't a carbon copy. Because US used imperial units for everything, Soviets simply couldn't make it a carbon copy because they could not e.g. source plating and wire of the exact right size. So they replaced it with the nearest metric equivalents that were available, erring on the side of making things thicker, to ensure structural integrity - which also made it a little bit heavier than the original. Even bigger changes were made - for example, Tupolev insisted on using existing Soviet engines (!), weapons, and radios in lieu of copying the American ones. It should be noted that Stalin really did want a carbon copy originally, and Tupolev had to fight his way on each one of those decisions.
That said, even then Tu-4 wasn't a carbon copy. Because US used imperial units for everything, Soviets simply couldn't make it a carbon copy because they could not e.g. source plating and wire of the exact right size. So they replaced it with the nearest metric equivalents that were available, erring on the side of making things thicker, to ensure structural integrity - which also made it a little bit heavier than the original. Even bigger changes were made - for example, Tupolev insisted on using existing Soviet engines (!), weapons, and radios in lieu of copying the American ones. It should be noted that Stalin really did want a carbon copy originally, and Tupolev had to fight his way on each one of those decisions.