Ironically, i had a colleague have this exact issue last week.
He was getting the same errors (i.e. the APED errors etc) , but underlying condition was that the token/license server wasn't getting configured through the SLM Config.
In the configuration process the registry was not getting updated with SLM server details, so we brute forced the actual registry keys required. HYSYS would then run, but the errors persisted.
Trying my suggestions above didn't show any fruit. As we were not able to select a database to get going.
One of our fall back plans to sort out the SLM, was to redo the install, but as administrator.
Our services guys, ended up taking that idea to try and sort out the APED errors, by running the "Aspen Properties Database Manager" as administrator and he was able select the database, which in turn cleared the error.
So in short, try running the "SLM Configuration Wizard" as administrator, this should allow the application to actually write to the registry
And run "Aspen Properties Database Manager" as administrator to select the properties database.
What i found interesting is that an error didn't really spit out from the SLM config program saying that it was unable to write to the registry, it didn't do its normal routine, but said it was successful (where you could see it wasn't)