As I understand it, all other things being equal, handedness is a spectrum thing not a one or the other. The spectrum ranges from very left handed through left handed only for most/ some/ a few things to ambidextrous and then by degrees to very right handed.
No they don't - and neither do right handed people do everything with their right hand. I'm right handed but do things like holding a snooker cue, dealing cards, or shooting a gun or an arrow from a bow with my left hand.
I write right handed, play cricket and golf left handed and kick a football with my left foot. I also write letters (individual letters that is) a different way to most people and didn't realise this until i was about 27 and my wife noticed.
No, I can't write etc, with my left hand, so I'm not really ambidextrous, Claymore - just 'caggy', as my mum used to say! :o)
It get's a bit awkward when I'm playing cards because I can't fan my 'hand' the right way and consequently I hold them upside down. It's okay if the card's value is printed in all four corners, but if it's only printed in two, I can't see what cards I'm holding. True. :o)
I seem to be 'naturally' right-handed but can do most things more or less equally with either hand except write, hammer and saw with left. To avoid RSI I swap mouse hands each month and get on fine.
Woofgang's answer makes sense to me. Being 'left-handed' I was forced to write with my right hand at school so my writing is not good. I do however remember when I started tech school in Holland I was gonna be a printers hand compositor which required you to hold a special hook in the left hand and put each letter in the hook with the right hand. It was some time before I realised that I was actually working like a right handed person but because I learned the task from scratch I was as good at it as others. There are(were) no left handed hooks, left handed people just learned to adept.
Eating with fork in left and knife in right has always come natural to me.
I am also very adept at using right-handed cups. (°o°)