Jenn,
This would be better answered by the Service Desk community and I have moved it over for focus. I can comment as follows though:
1. In theory it should, but Service Desk 12.6 was never certified beyond PAM version 4.1:
https://support.ca.com/phpdocs/0/8165/8165_r126_CertMatrix.pdf
2. If you would like to preserve your existing processes, and in flight process instances, you can move your PAM install from one server to another, then perform the upgrade.
You can move the 3.1 install and point it back at the existing database by following the tech doc:
Knowledge Doc: TEC596124 How to configure a new PAM Orchestrator instance to use existing PAM database tables
After that I think you would just need to modify Service Desk to point to the new PAM Orchestrator Server.