I can't think of a nice way to say this, so I'll just say it: it is highly unlikely your GT will ever be fixed gear. Vertical dropouts are a HUGE road-block for an SS/FG project. Buying parts on ebay and "seeing if it works" is another red flag. You will end up spending lots of money on this bike and not get what you want out of it.
Sell what you have left and buy a bike with horizontal dropouts if you really want to undertake an SS/FG project.