If I go back to block #1000 and add one more transaction, this will change the hash and it wont be the same one stored in #1001. However, how the hash of block #1000 changes in the first place if it was already calculated?
Once a block has been mined and accepted as part of the longest valid chain (ie, several confirmations deep), it will not ever change (which means the block's contents, and thus hash, will not change).
In order to change it, you would have to also change every block that comes after it, all the way up until the current chain tip. This would take a huge amount of computational power to pull off, you would basically need to 51% attack the network, to re-write the blockchain history. This sort of attack is extremely expensive, and ignores the incentives for mining honestly, you can read more about it here.
Otherwise, if you just changed block #1000, and then attempted to broadcast the changed block to your peers, they will all reject that block, since it is not part of the longest valid chain (assuming they have already heard about the original block #1000, and all of the blocks that came after it).
does this mean that the hash of each block is recalculated regularly?
A block's hash will be calculated in order to determine its validity, but we would never expect the hash to change for a block that has a sufficient number of confirmations on the network.
