I want the average velocity within the red selection box, not the average for the entire cross-section. How should I proceed with the selection?
Using Finite Slice Planes may be the best option. You end up with a slice plane covering the desired area - then use the Integration or Engineering Quantities tool on that reduced-size slice plane.
https://help.altair.com/hwdesktop/cfd/topics/post-processing/slice_planes_create_cfd_t.htm