I guess theoretically you could root without unlocking the bootloader, but it would probably require a substantial amount of hacking. The phone I had before the N5 was like that, it was an AT&T Galaxy S4, and the bootloader was locked down, but it was still rootable with a CF script. Of course this is moot seeing as the Nexus is easily unlockable.
Not if you use this:
Unlocking the bootloader the first time requires fastboot, however.
Nonetheless there are instances where this will not be an option. It is entirely conceivable to accidentally flash yourself into a corner where you have stock or otherwise unusable recovery, and a non-bootable system partition. Should this happen, the only way to save your data would be if you already have an unlocked bootloader so you can reflash custom recovery and get in at your data.