Do I need to remove remaining wallpaper backing before painting?
Those "brown spots" are the inside of your drywall's surface paper. You're actually pulling off the surface of the drywall (which is the gray you see adjacent to the brown). It's not a total disaster, but normally you'd use a little moisture to dissolve the adhesive and gently scrape it off. 

Since you're going to paint, you'll need to do a few things for a favorable outcome:


Finish cleaning up the wallpaper to a reasonable extent. It doesn't have to be perfect since you'll need to skim things anyway. Use a putty knife to scrape away any remaining loose paper. 
Roll on a drywall sealer or PVA primer. This will stabilize the exposed paper and provide a better substrate for repairs. Let that dry well.
Skim the walls with joint compound using a large taping knife or steel trowel. Apply several light applications as needed. Each should be very thin. You're not "spackling" or "plastering" here. You're just skimming. Think of it as applying a coat of paint only.
Sand lightly with a drywall screen or other large sanding block. Don't use a small block as it'll result in waviness. It's ok if some paper comes back through. Press lightly and use long strokes in multiple directions. If necessary, repeat steps 3 and 4.
Prime again and paint.