Do I need Self-Signed Certificates for TaroWorks to run?

No.

Self-signed certificates gets created after enabling the domain in the Salesforce development organization. As a part of TaroWorks package, we do not use any Self-Signed Certificates on our APIs for authentication purpose. Hence, disabling the certificate does not impact any of the TaroWorks functionality.

We also do not need to renew this certificate for TaroWorks, except in cases where there is any specific reason for using this certificate other than TaroWorks.

If you need this certificate, you could use a 3rd-party certificate instead of a self-signed certificate by sending the certificate request to the Salesforce certificate authority. After the certificate authority sends back the signed certificate, then you need to upload a signed certificate in the Salesforce organization.