The people on the Flash and Short Story forums could advise you better, but probably the best website for publications which take short stories is:
http://www.duotrope.com/
If the website where your story is currently uploaded is open to google searches then, yes, you should remove it before you submit it. (NB. WW forums are protected from google searches if the work is uploaded to be visible to WWmembers or Group Members Only, but may be visible to searches if uploaded in the Topic sections of the groups)
Your story will probably need some sort of modification to fit the various submission criteria of the different magazines - mostly word count, but also subject matter depending on what each magazine specialises in.
Magazines usually advise reading the stories they publish first before submitting so you get an idea of what they are looking for.
A lot of ezines (online magazines) don't pay for stories if they publish them, but it's all grist to your writers CV.
As for publishing history, everyone has to start somewhere so it's ok to leave this blank in your covering letter if you haven't had anything published to date.
Good luck.
- NaomiM