Do eeprom page writes always cost the same time as byte writes?
Yes, or "maybe".

The datasheet guarantees a maximum tWR of 5 ms, no matter what you write. That is, it specifically guarantees this for both byte and page writes:


  The write cycle time tWR is the time from a valid Stop condition of a write sequence to the end of the internal clear/write cycle


The actual time it takes can depend on quite a lot of things, and the only way you can find out if it is ready to accept another command is to perform the Acknowledge Polling procedure.