Are op-amps always assumed to be ideal?
No, depending on what you are trying to do you might have to use a more complex op-amp model with finite gain, input offset voltage, input bias current, etc.

If it is a real problem you can apply experience and judgment to decide what can be left out. For set problems the rule usually is that you should use all the information given. If nobody mentions offset voltage, assume it is zero. There are exceptions- sometimes they'll throw in an irrelevant number (find DC voltages and they tell you the Gain-bandwidth product, for example), and sometimes you might be expected to parametrize the unknown variable.