Will fooling around lead to real playing?  

If she enjoys "playing" then try to see if she can learn to really play. Pick a children's song and see if she can learn to play that. If she likes that then pick another. If she still wants to go further then look to a teacher to guide her further. Don't push her to be a child prodigy, just allow her to have fun and see where it goes.

It depends. All of my cousins loved to bang on the piano and make noise on it. I would sit and pick out melodies by ear (keep in mind I was 8, if your daughter is the one in your avatar, she looks significantly younger).

Now I play piano (not exceptionally well though, just for fun) and a bunch of other instruments (most at levels better than the piano).

None of my cousins really play anything. One tried learning violin for awhile and it was a horrible racket. She ended up breaking the instrument. But if you want to see if this piano "playing" leads to anything without driving yourself insane, try getting an inexpensive electric keyboard that will hook up to earphones. Then she can play and you won't hear it.

When I was little, I also drove my parents crazy by banging on their piano. I guess when a child pays a lot of attention to something it might be their way of saying that they are interested.
Try to take your child for piano lessons. That's what my parents did and today I am a professional pianist - I dont regret it for a second.
Studies have shown that musicians are also better with their school work, specially maths, because the RH side of their brains are more stimulated than normal children.
From the age of 6-7 is the most critical age to take your child for piano lessons, because that's when they learn the quickest.
To be honest, just "playing" around on the piano, won't lead to real playing, as there is so much to piano playing than people think: there are trillions types of techniques, and every one tells a different story.
So, try the lessons. If you see after a few years that her interest didn't really develop, then at least you can't say that you've let a possible opportunity pass by.
Hope this helps ;-)

Ohhh I wouldnt stop her at all!!if she loves to tinkle then maybe in a few years she may want to take lessons..you dont say how old that she is?but I would have loved my children to play the piano,and although my son was musical there are none who play the keyboard.My relatives used to play the piano,and the squeeze box,the accordions and the violin/fiddle,and we all used to get together and have an impromtu party,it was fabulous fun ad someting that I will never forget.So just be patient for a while and let her develop!!and you nevr know you may be in for geetting her some piano lessons soon!!good luck..