A profile photo is usually displayed small and often as a circle. Both facts should decide how you crop it.
Profile pictures are shown at a few dozen pixels in most places. A full length photo at that size is an unidentifiable smudge, which is why headshots are the convention. Fill the frame with your face.
Assume it will be cropped to a circle. Most platforms do this, and the corners of your square simply vanish, so anything important near them is lost. Leaving margin at the sides costs nothing and protects against it.
Skip the background removal. A plain office wall or a softly blurred background reads as normal and professional. A hard cutout pasted onto flat white reads as a passport photo, which is a different thing and not usually the impression wanted.
Crop square and leave margin. The platform will apply a circle if it wants one.
Around 800 pixels square is plenty and uploads quickly.
No.