If I create a rebus or pictogram & post it here, does it then "belong" to this site, i.e. would it be OK if I posted it at another site?
If you write something here, it's yours and you are welcome to post it elsewhere too.
Two things to bear in mind, though:
Anything posted here is licensed under the so-called "CC-BY-SA licence with attribution required". That doesn't reduce your right to post it elsewhere in whatever way you like, but it does mean that you don't get to stop other people doing likewise. [EDITED to add:] Provided, of course, that they do so in accordance with the terms of the licence, which in particular requires them to give appropriate attribution.
We do rather frown on using PSE as a promotional tool. Reproducing puzzles you've posted here elsewhere needn't be that (indeed, it needn't be promotional at all), but I can imagine some courses of action you might have in mind that would be problematic. (E.g., everything you post here is also on your site, and all your posts have links to that site, and the real point of your participation here is just to get more traffic to your site. That wouldn't be OK.)
But, to reiterate, if you create something and post it here, there's nothing at all wrong about posting it elsewhere too.