Why don't you feel mosquito bites when they happen?

Firstly, they're very light so unless you're highly attuned to their presence you won't feel them land on you. However, the reason you won't actually feel her bite is that they release a fast acting and local anaesthetic at the point where her proboscis penetrates your skin. It's not the bite that's important to her, it's getting the blood, so they need to numb the point of penetration so that you remain unaware while she sucks the blood out. The anaesthetic wears off after a few minutes and explains why you rarely catch them doing it and often wonder when you got bit when you suddenly start itching and notice the bite.

And yes, it is only female mosquitoes that bite and feed off animals or humans - they need the blood as food for their larvae.