PNG Logo to Vector Cleanup

I have a logo only available as a small PNG and I need it rebuilt in Adobe Illustrator as a fully-layered vector file. While you redraw it, please correct imperfections - there are four claws not three; the tail fans out in symmetrical order; the legs look defined; the circles are clean circles - every curve is perfectly smooth and all anchor points are clean.

I'll be relying on your designer aptitude to make any changes that would improve the logo - be it a minor tweak or balancing or an introduction of an element to it. Colours, typography, and general styling stay exactly as they are—unless you recommend these changes.

Deliverables
• An .AI file with logical, clearly-named layers (icon, text, background, etc.)

I’ll consider the job done when I can, toggle layers independently, and drop the file straight into print or web layouts without further edits.