No don't have to start with a freewheel, it would just be easier and I didn't want to see you get fed up and quit while trying to learn basic handling skills.
As far as the bars, get what makes you comfortable but get the riding down first. I have bullhorns on my SS and like them, two fixed with mustache and drop bars, the flat bar never gave me enough different places to put my hands. If you do decide to switch to a flat bar make sure to get the same size so your brake levers fit.
Are your brake levers on the end of the bullhorns or down by the stem?
I really like this set up: