I don't have access to my computer where I have the program, so forgive me if I am a little less detailed, but I wanted to mention that I've had similar results (a screen shot would help) and found that if I re-fit the eyes, I was able to eliminate the problem. When you fit the face you place tiny dots around the eyes. The placement is very important as if you're a little too close you can wind up with a black hole and if you're too far out you can wind up with a ring around the eyes. Play with the dots around the eyes and you'll probably solve the problem.
Now it's important to note when you're doing so that when you start out, there are four dots around each eye. This is for general placement. If you're having problems I suggest pressing on the icon for fine control. That will give you around 8 control points for each eye, allowing the fine control that will probably solve your problem. The fine control icon for the eyes I'm referring to is found in the upper portion of the face fitting screen. It will look like an eye with a bunch of dots around it. To reach the face fitting screen on a picture you've already fitted, just go into the character creator pressing the 1st stick figure icon to the left of the main window, then you'll see icons for refitting the head (select it if you don't see it) in the form of eyes, mouth etc. Just fit the eyes.
I'm sorry it took so long for you to get a response. I just noticed this. Without a screenshot it's a little harder, but if you still have problems after following my instructions, I could take a look at your file and see if I could fix the problem if you'd like.
Now as far as moving the eyes, the mouth and expressions, there are a couple of ways to do that. My favorite is the puppet editor. It's actually kind of fun in fact. In the main screen you should see a couple of face icons on left, in the lower portion of the tool bar. One shows just a face. That allows you to change expressions using numeric information for different regions of the face. Not far above that is another icon of a face with a mouse next to it. This is the face puppet editor. It allows you to change expressions by moving the mouse, close and open the eyes by clicking the mouse buttons, etc. It allows you to preview results so you can see what will happen, before actually recording the action. Some actions may be too extreme from some of the options, creating the type of eyes you stated too. There is a slider to exaggerate or unexaggerate the motions that you can move until the results look perfect.
You can find a copy of the manual to assist you with further details here: http://ns.reallusion.com/help/eng/cta/CrazyTalk_Animator3_Manual.pdf
or ask for more help. Your post remained unanswered for a long time, but usually you will receive a response much quicker.