The reason I’m asking you about what generated the request is because if PMS is running native (the SPK installed in Package Center),  it is a native application.
This single line:
May 19, 2018 21:06:54.500 [0x7fc794847700] DEBUG - Filesystem Type: configfs Path: /config Name: none
is the disturbing part.     /config only exists within the confines of a Docker container.     Docker-Plex knows to ignore this.   Regular PMS does not.
As I look further,   I find this:
bytes (pipelined: 1430)
May 19, 2018 20:42:14.530 [0x7fc7f0d11700] DEBUG - handleStreamWrite code 110: Connection timed out
May 19, 2018 20:42:14.530 [0x7fc7f0d11700] DEBUG - Removed transcode output stream 0x7fc7eb4c1000, active count 52 => 51
May 19, 2018 20:42:14.537 [0x7fc7f0fff700] DEBUG - handleStreamWrite code 110: Connection timed out
May 19, 2018 20:42:14.537 [0x7fc7f0fff700] DEBUG - Removed transcode output stream 0x7fc7b4ce4000, active count 51 => 50
May 19, 2018 20:42:15.053 [0x7fc789bff700] ERROR - Throttle: timed out trying to read chunk 38
May 19, 2018 20:42:15.116 [0x7fc789bff700] DEBUG - [TranscodeOutputStream] Input processing thread exited after writing 3266 bytes, m_closed=1, m_endOfFileReached=0, session->isStopped()=0
May 19, 2018 20:42:15.117 [0x7fc7f0d11700] DEBUG - Auth: authenticated user 1 as iFredOS
May 19, 2018 20:42:15.117 [0x7fc7bde23700] DEBUG - Request: [192.168.168.51:61567 (Subnet)] GET /library/metadata/2262/art/1526765654 (72 live) GZIP Signed-in Token (iFredOS)
May 19, 2018 20:42:15.119 [0x7fc7f0d11700] DEBUG - handleStreamWrite code 110: Connection timed out
May 19, 2018 20:42:15.119 [0x7fc7f0d11700] DEBUG - Removed transcode output stream 0x7fc7eafe2000, active count 50 => 49
May 19, 2018 20:42:15.122 [0x7fc7bde23700] DEBUG - Computed media url for item 2262: http://127.0.0.1:41810/system/agents/media/get?guid=com%2Eplexapp%2Eagents%2Eimdb%3A%2F%2Ftt1306980%3Flang%3Den&mediaType=1&url=metadata%3A%2F%2Fart%2Fcom%2Eplexapp%2Eagents%2Eimdb_57f0b4e7fed221fb9fdb0721800432c577435469
May 19, 2018 20:42:15.122 [0x7fc7bde23700] DEBUG - HTTP requesting GET http://127.0.0.1:41810/system/agents/media/get?guid=com%2Eplexapp%2Eagents%2Eimdb%3A%2F%2Ftt1306980%3Flang%3Den&mediaType=1&url=metadata%3A%2F%2Fart%2Fcom%2Eplexapp%2Eagents%2Eimdb_57f0b4e7fed221fb9fdb0721800432c577435469
This is telling me PMS can’t write to the connection to send transcoded output.    This is indicative of a network connection problem.
Since I’m not seeing obvious  errors;  Do you use Jumbo ports in any way / adjust the MTU to anything other than 1500?   Something is slowing it down.
The transcoder’s inability to send,  coupled with your inability to get premium metadata from the internet, coupled with the initial error of the Syno not being able to reach the internet at all,   really looks like a fundamental networking problem but I am not seeing it.
Would you please elaborate on your DMZ configuration as well as how you have the Diskstation’s network configured?