Do all mosquitoes feed on blood?

No, only the female mosquitos feed on blood.

Mosquitoes primarily feed on nectar from flowers. Both males and females will feed in this manner, sucking the nectar through a proboscis, much the same as a butterfly does. However, the female is also capable of drinking blood, an act called haematophagy. Females do not require blood to survive, but they need supplements, such as protein and iron, to enable them to develop and lay their eggs. The female Mosquitoes do not actually "bite". They will actually pierce the skin with their sharp proboscis, injecting a mild painkiller as they do so, to numb the pain, and will then proceed to suck the host's blood.