Should you place traces at right angles through a via?
"because it can cause problems during manufacturing"  


If right angle traces come loose you have to go to another PCB manufacturer. Quality PCBs don't have this problem anymore.  

On HF boards right angles are avoided because of radiation they cause (reference, p.14). Otherwise the only reason seems to be aesthetical. You can safely part from your via on the other side of the board any direction you want.  

If yours is an HF board, keep in mind that vias cause extra inductance and capacitance, and cause reflections because they change the characteristic impedance.