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.
A logo is mostly flat colour with hard edges, which suits the hard edged brush well. Use a large brush to clear the white area quickly, then a small one near the letters. Put it back restores anything you took off by mistake.
Begin with the largest, cleanest version of the logo you have. The working copy is limited to 1400 pixels on the longest side, which is plenty for most logos, and the result is a PNG with a transparent background that you can place over any colour.
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.
Only if it is larger than 1400 pixels on its longest side, because that is the size of the working copy. Smaller logos are saved at their own size.