Does replacing a damaged SIM slot changes the IMEI as well?
The SIM slot is independant from your antenna/radio so your IMEI will remain the same.
No need to worry about it at all.

The IMEI itself is usually inside your phone's internal read-only memory (On Samsung phones it seems to be inside the /efs folder).
Sidenote: Your IMEI + your IMSI are unique identifiers for your device. It's similar to a MAC address but works for cellular networks. Every IMEI can only connect once to every transmission tower, two devices with the same IMEI would cause both to lose connection.