Transient analysis is what you need to use, not superposition. Superpositions is when different frequency signals are in the same circuit and you analyze the circuit separately for each frequency.
For initial conditions (t<0) you assume that the switch has been closed for a very long time and so C1 is storing no charge. Then you do transient analysis.
No, if it were so, the solution would be trivial. You have to assume that IB1*R1 is not equal to Vos. But v+ and v- of the opamp do have to be equal. Where does the excess current go?