Hi Mark,
Same here mark 
Please note that Source and target databases both are running on ASE version of 15.7.
The log full issue was not present in ASE 15.5(earlier production system..now we migrated to 15.7 completely) as truncate log on checkpoint was enable for this databases. We are not making this DB option enable in 15.7 as it's a production environment..
1.Are you having problems with the log filling up in ASE 15.5? [yes or no] ---> 15.5 version system is not live currently so it doesn't matter.
2.Are you having problems with the log filling up in ASE 15.7? [yes or no] --> Yes
3.3a - if the database is a primary database (PDB) in a replication configuration, is there a matching record in syslogshold? ---> Yes
3b.This databases are getting replicated to the target server(databases which is shown in syslogshold)
So that is the reason i think entry for this databases in syslogshold is a valid.
3c.if the database is not part of a replication configuration, is there a matching record in syslogshold?----> No such records for db which is not part of replication is present in syslogshold.
3d - when the log is 80% full (or any appreciable amount below 100% full), does 'dump tran /to_file' or 'dump tran w/ truncate_only' clear the log? --> No, both of the above two dump tran option was not successful to clear the log.
3e - Is the log 80% full, or (near) 100% full, when you say that 'dump tran w/no_log' is the only command that clears the log? -Yes
3f - Once you've issued 'dump tran w/no_log', does log space usage drop back to/near 0%? alternatively, what does 'sp_helpsegment logsegment' show after running 'dump tran w/no_log'
---> I didn't remember the exact values, but yes the log space usages values was less than 3-4 %.
Let me know if any more information is needed from my end .