A logo with a white box around it looks wrong on every background except white. Cutting the background out fixes it everywhere at once.
Save the finished logo as a JPG and the transparent area comes back as solid white, putting you exactly where you started. PNG is the format that stores transparency, and WebP does too if you need something smaller.
Start from the biggest copy you can find. A logo cut out from a small, blurry image stays small and blurry, and enlarging afterwards makes the ragged edges worse rather than better.
Always check on a dark background. Pale fringing left around the edges is invisible on white and glaringly obvious on black, and it is the most common reason a cutout looks amateurish.
It was almost certainly saved as a JPG. Save as PNG instead.
You can, but the result inherits the limitations of the source. Use the largest version available.
No.