My host provides PHP version 5.5.30; my moodle version is Moodle 2.9.6 (Build: 20160509) - I think this is OK. The environment check in moodle confirms min requirements are met - but it does mention the barracuda format (which is unavailable to me).
Regarding the backups - I just upgraded to the above build from 2.9.5 and have a backup of one course. The other course (only running two currently) is the one I cannot seem to backup.
The moodledata/temp/backup file has many 4KB and 0KB files (failed backups?) I'm not sure if one of those is the file you are offering to examine.
The backup failure usually occurs at around the 96.4% point.
If you think there is some benefit to examine one of these I can make it available - let me know; I have not used PM but will watch for messages.
Really appreciate your interest - wish my tech skills were worthy!