louise
you sound so like me, i spent most of my little girls' toddler years trying to get things right for her, trying to be the most patient, understanding mother (for which read SLAVE) possible, diffusing potential rows and tantrums, and suppressing (for a time) enormous feelings of anger, regret (missing my old life SO badly!!) and bewilderment ('surely it isn't supposed to be like this?!')
IMO 3 is the very hardest year (so far - my oldest is 7 now, i realise i have the eek teenage years to come!) esp when they are girls as others have said! at that stage i really was a slave to my 3 yr-old ( i look back in horror at how much she dominated our family life and most especially mine) for all the reasons you feel you are at times. i wasn't so much depresseed as stressed (insomnia, weight loss, emotionally labile) not helped by being in australia that year, and far from any support network. in fact, i used to fantasise about having a small (not large or excessively painful) road accident, just enough to put me in hospital for a few weeks of RESPITE!! so please believe you are not alone with that feeling of looking back at your old life with longing and, at times, regret, i have definitely been there.
playdates for me at that stage were hard as my daughter is quite introverted and was not great at socialising. i too used to get roped into imaginary games for hours at a time while washing piled!! argh!! however slowly but surely things have got loads better. i found having a sibling v helpful for her, even though life was SO HARD for a couple of years combinging willful toddler with a (gorgeous! but still...) baby. but now, she has a brother two years younger who is her best friend. he's much more mellow than her. he is a master of diplomacy and knows how to handle her moods and tantrums as well as we do! play dates became loads easier from the age of 5 and she has lots now.
playing alone is a great skill to teach your child. i have to say that i was rubbish at imparting this lesson and just lucky i have two who get on with each other (most of the time). i think others have given good advice on this one and i wish i had been a mumsnetter back then!
good luck, you are definitely not alone with these feelings.