I got the cog and lockring on as tight as I can, but under super hard backpressure, like slamming backwards, almost standing on the crank, which is not something I one would do in normal riding, the cog still slips. It moves the cranks less than 1/8th of a turn, but it still makes me uneasy. Should I be worried, and is there a way to fix the problem? I mean, it won't even move during a skid or skip.

Yes, you should be worried. If it's not already, your hub will soon be be gargabe. It may be that you installed them poorly, or a quirk of the cog / lockring combo you are using. What brands are you using? Sometimes they don't play well together.
Or maybe it's nothing to worry about. Who can tell.

Is it the old surley cog? They tend to be thinner than other cogs allowing them to move. You can use a spacer or get a new cog.

I think I'll just get a spacer of some kind. It's really not much play, but enough. Thanks.

Is it a converted road hub or a "real" track/fixed gear hub? My normal method for cog/lockring installation on track hubs is to put a little anti-sieze on the threads, thread on the cog and tighten with a chainwip, thread on the lockring and tighten with a spanner, then take it out for a brief ride, using only the brake to stop (or my feet if there isn't a front brake) while not applying any backpressure. After a few minutes of pseudo-sprinting, the cog will be tightened as tight as it will ever go, I think crank the lockring down again and the setup is good to go.

I've found that if I skip the "riding around" step, I can get some play in the cog/lockring combo that will allow my cog to thread and unthread a little bit. I like the idea of jamming a spacer in there a little more than this, but a little less than just getting the parts you have to interact properly.

yes, its a real track hub. I cut a thick plastic spacer out of a milk jug and rode it around, its better, but not perfect. I think tonight I'll try that method, after a fast sprint home.

Just make sure to use your brake to stop (or a foot on the ground after "supermaning" for a bit if you don't have a brake) because applying backpressure by the pedals kind of ruins the whole sprinting-around-to-get-the-cog-screwed-on-tightly method.

Yeah, I gathered. Thanks for everyones help, I did a hard sprint while holding the brake, and while I most likely need new pads, I secured the cog firmly, and the lockring plus milk-jug spacer is on there tight as a drum. To test it, I got going, then pulled back, hard, with both legs, putting most of my weight on the back, WAYYY beyond any street move, and nothing happened except some spoke-settling sounds. Thanks again!