Is there a way to remove the eye blinking animation on a character without opening up an animation editor?

Secondly, is there a way to remove the pupil that appears over the eye textures of character models, and how might that be accomplished?

Well, I think the easiest way - to change a character 3D model. It is necessary to remove eyelids and pupils.

Technically, you should do something like this:
http://forums.runicgames.com/viewtopic.php?f=47&t=60172&p=496180&hilit=scar#p496180
This scar more difficult task, but the same principle.

Thanks for getting me started! It will take me a while to get serious about learning animation and mesh making, but its nice to know how it works when I finally get to that point.

Right now, I'm still acquainting myself with GUTS and the way code works in TL2, but as my project progresses, I have a feeling I won't be able to stop myself from delving into animation and mesh making. Or at the least, getting my feet wet.   

By the way, I just got done checking out the work you've posted - what you do is absolutely amazing! The Faceless are great - not to mention the terminator knight or the deployable cannon on the predator! Just. Wow.

I would be honored to have a model from you - though I don't think I'm worthy of it yet!
All I have to show so far are two working custom missile skills ("Internal Mounted Wrist Bolters" and "Bladed Disc Launcher"), three passives, and a half finished shoulder mounted laser cannon skill. (I've got all the particles and layouts done, now I just have to put them together.)

I've got a long way to go to fill out all the skill trees, but its getting easier, and slightly faster, as I'm learning more.

If I'm not mistaken, a "blink" is part of an animation that pulls some mesh-part over the eye-part for a split-second. That mesh-part has to be calling a part of the texture file that is all flesh-colored.

If you can find that part on the texture file and revise it so it's the same as the eye-part, it should look like he didn't blink at all.

@Kva3imoda am I right on this?

For Immortals I removed all unnecessary -  eyelids and pupils, also I removed a movable jaw. Now a face motionless as a mask. Change animations is no need, everything works fine without it.

I wondered if there was a way to do it with code after looking at the wardrobes file, where it says something about "add eyes" and "remove eyes."

This will not work for standard models. Because the wardrobe "eyes" is a certain area of a head, which is equal to half of the face.