I know you don't have to make reservations months in advance like at WDW, but if I call three weeks in advance for mid-week in July, will I be likely to get what I want?
You should be fine - I think the only thing you might have to book really early is the Princess character meal at Ariel's Grotto. Three weeks in advance is a long time here in CA!
I just made my PS ressies today. We are 50 days out. Mind you that we are going with another family (total of 8 people). I was not able to get the PS for BB or Ariel's that we wanted. I ended up have to go with what was available during our stay.
I have to agree with the previous poster-we are going next month and I called at the 60 day mark and still didn't get the times we wanted for our party of 7. We still got all the AR's we wanted, just on different days and times than we'd hoped. My advice would be not to wait until 3 weeks before, call at the 60 day mark.