Is a suspension fork with a lockout essentially a rigid fork or is there still some play?

no, it is not. even when you have "locked out" your fork, when you push on it, it will move. same with pro pedal. even when it's on 3, it will give if you put enough pressure on it.

It has some play, but very little.

My SID has very little play. Not noticeable when your riding. I think most lockout forks have a blowout valves for big hits,too . Great when you forgot to unlock the fork. Now is a lockout necessary is another question. If you sit for most of your climbs, you probably don't need it. But I stand a lot, riding a singlespeed, so I like locking it out.

Not to mention the fact that a thing with 2 ~1" stanchions is going to act nothing like a rigid fork.

I have a nice Fox and it helps to lockout, but as 20grit says, the fork tubes move more than a ridgid. Even when its locked out there is some up and down.

Well, that's the whole point.

But, to avoid damage in case of hard impact some movement (in my Fox amount is adjustable) remains.

No your still carrying 1kg + weight on the front end you dont need.
Get a proper rigid fork.

Any suggestions as to a proper rigid fork? I was just looking at the Surly Karate Monkey fork.

I think I am going to just get a rigid fork and see how I like it.

You first need to know the correct axle to crown length and rake of the original fork for that bike. Getting these wrong means changing your steering geometry. A Karate Monkey fork is a known stiff riding fork, they changed it to a better riding fork for this year ao I would look elsewhere IMO.