What happens when you make a free app paid? My app has always(4 months) been free and I am wondering if I make it paid, will it be in the new paid apps category?

It is not new in sense of app but new in sense of paid app.
That's up to Apple.  They decide what apps to feature.  The App Store gets thousands of new/updated apps every day and not every new app will be featured.  If your app provides some special service or interesting presentation, it's possible your app will be featured.