Why is a gibbous moon possible? This could be a silly question to ask, but I would like to know why is a gibbous moon shape possible.

From the few cases I've drawn here, I could not think of a configuration where the Earth's shadow can be cast on the Moon in a way that leads to a gibbous shape of the bright part of Moon. In my diagrams, grey represents bright and black represents shadowed portions of the Moon (smaller circles, with the bigger circles being Earth).
As for the order of the celestial bodies, the Sun is out of the screen, Earth is on the screen and the Moon is inside the screen. That could be a crude arrangement but it's the only one I can come up with to cast shadows, though I can accept the fact that there could be more complex 3-D arrangements.

 A: The shadow in a gibbous Moon is not caused by the Earth's shadow ─ Earth plays no role in that arrangement, beyond being the place where the observer sits. Instead, the shadow in a gibbous Moon is simply cast by the Moon itself. The correct arrangement is as follows:

(With due apologies for the poor drawing quality.)
This means, in particular, that your depiction of how the gibbous Moon looks like is incorrect: the terminator (i.e. the line between the illuminated and dark sides) is a meridian, and it passes (at least, modulo inclinations and so on) through the North and South poles. (This is true at all times, regardless of whether the Moon is a crescent or gibbous.)

Compare these two with a photograph of a gibbous Moon:

Image source
(Note, in particular, the dark side of the Moon, illuminated by Earthshine.)
A: The phases of the Moon are not caused by Earth casting a shadow. The Moon is only lit on half of its surface. As it revolves around the Earth, we see different amounts of that lit half. 
To see it yourself, take a ping pong ball or something and color half of it. Now rotate the ball while looking at it from the side, and you can see all of the phases.
A: The earth is not casting a shodow on the moon when it is gibbous. Shadows only get cast during an eclipse of the moon --- and this is rarish phenomenon. We have a gibbous moon every month.  In the absence of an eclipse, the sun  illuminates one half of the moon  leaving the other half in darkness. The moon appear  gibbous  when the moon is closer to the sun than the earth is. The sun is then mostly illuminating the side of the moon that we cannot see (the "far side") and  most of the side we can see is in darkness.  
