Glucas, Indeed most people leave, flee! during summer (that was also common in Houston were we lived too!) but there are many that decided to stay. This is our first time here and although we will leave July and August, I am sure there are plenty of activities for you and your kid. You can always check expatwoman.com and expatgossip.com, it won't take you a week to find the right company!
I have met already nice people there and most of the moms have kids in the age of yours. I cannot join many things because my son goes to preK and other activities and we live 'far' from most of the action spots :) but keep posting here and we might meet you. I know you are cautious with the email identity (me too, but rather because of my comments here about my frustrations in Qatar) but you will soon discover this is a very safe country.
All the best and do keep in touch.