in at-speed testing, launch will happen at hips frequency, but capture will be at slower rate. at-speed testing will give more about transitions, rise, fall and other speed effecting defects.
yes. at-speed test means test the design at its system speed.
Bist test for memory of IP blocks are usally at-speed (they will design to run at system clk speed).
For scan, the at-speed test happens with at-speed launch and capture during scan_enable pin low.