I've read somewhere that putting your story, or a piece of it, on the internet would make it difficult (not to say impossible in some cases) to find a publisher wanting to publish your script. Would I encounter such difficulties if say I let my story be bound in a book as a present to somebody (so only one exemplar), and decided later I wanted this story to be published?
Having a copy bound, either from a Do-it-yourself kit or through a place like Kinkos doesn't pose any problems. The problem with putting a book on the internet or by using a self-publisher or PoD publisher lies in the contract and the rights. By making a single copy and not actively distributing it or signing any contract, you haven't used any publication rights. When you sign on with a self-publisher or PoD publisher you sign a legal contract (even if it's just by pressing that you accept their ToS) and you effectively use up your first publication rights.
Also, you're allowed to post an excerpt of your story on your webpage. As long as it's not a significant portion of the story, you're okay. "A significant portion" does vary by publisher and there is no set "rule" for how much that is, but as long as you stay under 50% you shouldn't have any problems.
I will defer to some of the editorial folks, who may know better, but I believe you'd be okay. If you head on down to the local Kinkos and get your work bound nicely, nobody will ever know. The main issue with rights is if a work has been made "publicly" available. On a webpage, anyone with a computer can see it. A single copy in the hands of a single person isn't a big deal.
Self-publishing and POD packages, however, often include getting an ISBN for your work, and that will probably raise some flags.