Yes, you can reuse your old SSID.
Yes, you can reuse your old SSID.
Yes, you can use the same SSID and password after replacing your router. The new SSID will typically appear as a variation of the original, such as Network-2414 or similar, depending on how your device interprets it. Everything should connect automatically once you enter the correct credentials.
The method for setting up the SSID differs based on the router model.