Shape of urea molecule around N atom
I was supposed to predict the shape of the urea molecule $[\ce{H2N(CO)NH2}]$ using Lewis and VSEPR.


Don't forget that there are assumptions in VSPER to simplify the model. From Wikipedia:


  Valence shell electron pair repulsion (VSEPR) theory is a model used in chemistry to predict the geometry of individual molecules...


Essentially VSPER is for molecules in the gas phase where the only electrostatic interactions are within the molecule. Once the molecule is in a condensed phase, the assumption that it does not interact with external forces breaks down.


  What causes this change in geometry when the molecule crystallizes?


For one let's not forget that urea is a resonant structure, and does not have fixed bonds which also goes against assumptions of VSPER.



Further you have neighboring molecules with the hydrogens of those molecule taking some electron density from the resonance structure which is shown in the unit cell representation shown below:



In summary you have resonance which will flatten out the molecule and other molecules taking some electron density that combined will contribute further causing the molecule to flatten out.


  Does the same thing go for other molecules?


Yes, other molecules can condense and form structures that deviate from VSPER. Water in ice for example has a $109.5 ^\circ$ bond angle (tetrahedral angle) where as VSPER would predict $104.5^\circ$ (bent angle). 


  Would there have been a way of predicting this as I was solving the question?


No, you would have to know how urea crystallizes and many other factors, then used a computer simulation to calculate (estimate) such.