When you move the faces your not affecting where the projection goes. The center of the object is still where it was. The object just gets more empty space. When you move it as an object in that mode your change the locater and translates. This goes into the UV projection which was set for the initial location. This is all useless to you. What you want to know is how do I make a UV map and set up UVs and then apply textures. Im not a smart ass I wont post a tutorial here to make a point. There are great ones walking you through things like stools and pop cans. When your done those theres ones for characters. When your done those theres ones to streamline how you do it. Then theres ones for using 3rd party apps to further speed it up. By the time your done this you'll have opened your Christmas presents celebrated the new year started school in the next semester and learned why what the Hell Im saying. Welcome to Maya the high end package of choice for professionals the world round.

***EDIT***
someone should sticky the answer to the original question its asked more than how to render a movie.