How do I mount an external disk drive formatted in ext4 via USB Host? I plugged a 2.5 inch 1Tb drive into a Nexus 10 and I cannot see it anywhere really. Something pops up in ES File Explorer, but I cannot access the drive.

Do I have to root my device, or is there some other way?  Would formatting it to a different filesystem help in any way (to access it for r/w)?
You have a couple of options. The easiest way is to buy nexus media importer - it was $4 last I checked. You can use the nexus photo viewer app to make sure it will work. I used it with a flash drive on my nexus 10 and it worked great.

The alternative is to root and use stickmount. I prefer to root my devices, so I would recommend that way, but is does come with some security issues. It is pretty straightforward on the nexus 10, but if you are uncomfortable with the idea, nexus media importer is worth the $4.

Note that that drives like that need to be externally powered or they will not work - I don't think tablets can put out enough juice by themselves. 

If you would like to root I'd be happy to walk you through it, but it'll probably take enough time that it is worth just buying the app, unless you are interested in learning more about this kind of stuff.