Does cropping reduce the quality of my photo?

The pixels you keep are copied at their original resolution — nothing is scaled down. A JPG is re-encoded once at high quality, which is visually lossless; PNG and WebP are re-encoded losslessly.

Which aspect ratio do I need for a passport or exam photo?

3.5:4.5 — that is the 3.5 cm × 4.5 cm proportion nearly every Indian ID and exam photo uses. If you need an exact pixel size and KB limit as well, use the Exam Photo Resizer instead: it crops and hits the portal's spec in one step.