the black king?
Problems

As well as, if I hit refresh while the time is running it becomes my move. I don't see that as a connection issue.

Which browser and version are you using?
What type of internet connection do you have ? (adsl, dial up, wifi ? )
Do you have just live chess window/tab open when playing? or more ?
When was the last time you cleared cookies and history from your computer?
http://www.chess.com/clear_cookies.html
Is the computer you're using, work, public or home?

Safari 5.0.3
Wifi
The tabs vary however I don't flip between them.
I clear the cookies regularly, about a week ago was the last time.
This is my personal computer, a mac laptop.

I just cleared the cookies and loaded up the tactics trainer.
Here is the results:
As you can see, no black king is visible but he is there on f7.

that is odd. somehow your browser cache is causing issues. do you have any odd firewall or other program? try hitting control+f5 on that same screen when that happens. (btw, it isn't cookies that are the issue, but the cache of files.)
as for disconnects, if your wifi isn't super stable, that might cause problems.
I can't help but to be disappointed. When I first came to this site there were a few problems here and there but now it's just out of control. I routinely sit and watch my opponents time run out only to find that I lose because I supposedly abandoned the game. I can only see the black king in live games but not online or tactics trainer modes which sucks. I have updated my browsers and somedays can see things and have no problems while other days I can't. If anyone has an explanation that would be great.