Re: Nexus 6 Rooted OTA update
Yes, flashing is done via fastboot.
These are the commands that need to be executed in sequence. Execute one, wait for successful completion, then the next and so on. These steps below assume that fastboot and the img files are in the current directory (or that fastboot command is in the system path variable, and that the current directory has the IMG files). If not, you'll need to enter the full path of the IMG files
fastboot flash bootloader bootloader.img
fastboot reboot-bootloader
fastboot flash radio radio.img
fastboot reboot-bootloader
fastboot flash system system.img
fastboot flash boot boot.img
fastboot flash recovery recovery.img
fastboot flash cache cache.img
To give credit where it is due, and for further reading before you do any of this (assuming you haven't done this in the past), read
[TUTORIAL][GUIDE][HOW TO] Flash Factory Imag… | Nexus 6 | XDA Forums
Remember that the thread I linked to is for restoring to stock, so it has a step to flash userdata.img (that will wipe your data), and it also has a step for factory data reset - which will also wipe your data. So, skip those steps if you want your data retained. And if your bootloader isn't unlocked, unlocking WILL wipe your data.
There are two methods described there. Use Method 2 (that's what I described above)
No - you can't flash factory images using TWRP. You can look thru' XDA and see if there is a flashable image available from some dev. I don't know.