Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 21 / b

All the listed modes are Raman active; only $A_1$ and $E$ are infrared active. For the C4v molecular point group, with vertical planes through the outer atoms:

There is no inversion centre, so infrared and Raman mutual exclusion does not apply.

In the stipulated ideal, fully resolved limit, yes: square-planar geometry gives three infrared fundamentals, whereas square-pyramidal geometry gives four: two parallel $A_1$ bands and two perpendicular $E$ bands. The extra parallel fundamental is the distinguishing feature. In a real incomplete spectrum, a weak band, overlap or accidental degeneracy could hide that difference, so absence of a fourth observed band would not alone prove planarity.