Whether the leadfield matrix is fixed

No, the geometry of the head is important for modelling accurately the currents. If you have individual head shapes and electrodes positions, the leadfield is computed again for each spatial configuration, it does not use any generic head model.

If you want, or if you don't have individual anatomical scans or electrode positions, you can use the same head model (MNI ICMB152 template or other) and the same electrodes positions. But expect a lower spatial accuracy.