How much space an ascii character really takes on a 64 bit word addressable memory? I know that an ASCII character needs 1 byte of memory for storage, but if a computer uses a 64-bit word addressable memory does it mean that the character actually takes 8 bytes even when only 1 byte was needed?
Does being a 64-Bit word addressable memory mean the same as being a 64-Bit machine?
It depends.  A program can store 1 ASCII character in each 64-bit word, or 8 ASCII characters in each 64-bit word.  It's up to each individual program to decide how it wants to store and format its data in memory.  The latter would probably be more typical.