What the general health checks to be performed Pre and Post patching the SharePoint servers for SP 2013?
Are there any recommended best practices?
I can share some steps from my experience.
All above test will tell you the actual picture of your farm.
Things I check before upgrading to a newer patch:
Post patching, after running Product Configuration Wizard I check Central Admin: Check product and patch installation status, and that all web applications can be accessed from a client machine.