Since you mention you have a sinusoidal load (specific frequency or a range of freqeuencies) you can probably do a harmonic analysis instead of the transient unless you are interested in transient effects at the beginning of the load or end of the load. The harmonic analysis will give the steady state response of the structure under sinusoidal load at a particular frequency.

The way you have set up the modal analysis will be based on the deformed configuration from the static analysis and will take into account the stress stiffening effects due to the stress in the static analysis. Then the subsequent mode superposition transient/harmonic analysis will use these modes. However the thing to NOTE is that the stresses from the static analysis are not included in the transient or harmonic results. Only the mode shapes are used. So you may need to add the static results to the transient results or harmonic results separately using solution combination.

The resonant conditions occur when the applied force's frequency matches the natural frequencies of the structure and if the mode shape has components in the direction of the applied force. This will be obvious in a harmonic analysis when you plot the structures response vs frequency. In a transient this may not be obvious but is taken into account. You can see it by comparing to a static analysis with that load or with differnet amounts of damping.