The thread is long and I have not read all of it as I am supposed to be working, so apologies if I say something others have already said.
3 is in my opinion very young for going on playdates at peoples homes, there is bound to be tension at some point between the kids, and if the mums dont know eachother, or the other child well, it is difficult sometimes to know how to handle it.
It was not until DS1 started school that I started inviting other children home for playdates for the sake of DS1 rather than me. Up til then I only asked friends with children similar age to come over, as much for me to have a natter as for my DS1 to play.
Same with DS2, who is 2 1/2, I have never invited a child over, but I have asked friends with kids to come over.
Better than a playdate at home I find, is to ask to meet up in the park, at the playground, at the soft play, or a toddler group as this is neutral territory.
I think you are not doing anything wrong. It is hard when you are new to an area, you dont already have a mix of friends with kids locally, you have to start totally afresh.
I think that other mum is wrong to give her dd so much power, she is letting the dd take control over the situation, which she is far too young for. A 3 year old is short sighted, and knows nothing of cause and effect and consequences, a 3 year old thinks short term. It might have sounded a good idea at the time for the dd to get your ds out of the house so she could play with her toy, but it is a rubbish approach. At that age children should be taught to share and play together, not to kick their friends out rather than negotiate a situation.
Are there any groups in the area you can join with your DS? Monkey Music, Kiddie Kicks, or some other activity where you get to meet and get to know other parents?
Mumsnet meet ups?
Good Luck