Measured weight:

Raspberry Pi Zero + SD card 8.8 grams
Raspberry Pi Zero + SD card + pin header 11.8 grams
Raspberry Pi Zero + SD card + USB wlan adapter (Asus USB-N10 without covers) 11.9 grams

Measured power consumption at 3.3V fed directly to header:

150mA without USB devices
350mA with Asus N10 during ping in weak network connection