IP address over ethernet is self-assigned when using a gigabit adapter

The extensive troubleshooting you performed seems to indicate the problem lies somewhere between the CAT5e socket at the wall in the office and the router. I realize the iMac was able to connect using that same office socket, but only without the Belkin adapter. Introducing the Belkin adapter causes it to fail. You know the adapter itself is ok though, because you can successfully use it when you connect directly to the router in the living room.

If my understanding is correct, the explanation may be that the Belkin adapter is unable to negotiate a connection through that office socket and the router. What that suggests is a Cat5e connection that is not actually meeting the rigorous Cat5e standards. The Belkin adapter itself may be ok but the connection isn't good enough.

The fact the TP-Link switch causes the same failure may be another indication of that substandard office connection. You didn't say whether it was involved in it or not though, or if you tried a different cable.

Lacking some independent testing equipment, the most expedient way of confirming my suspicion is to completely bypass that office socket with a sufficiently long Ethernet cable. Gigabit Ethernet is unforgiving of substandard connectors or wiring. Not enough twists per inch, or a broken / intermittent connection, and things won't work.

Try bypassing that office Ethernet port. I think you will find the culprit will be in the office connections or cables you had been using, and your challenge is to isolate the cause.

The following is paraphrased from my notes regarding a discussion I had with an Apple engineer several years ago. The subject of the discussion was the amount of time it took to establish a fast and reliable Ethernet connection. It was taking on the order of 30 seconds or so, and I was wondering why it took so long to establish compared to a comparably fast wireless connection.

"The minimum protocol required to enable an Ethernet connection is 10Base-T.

"When you connect a 1000Base-T (1 GigE) switch, adapter, or other device, the protocol that communicates the Speed / Duplex / Flow Control between the devices is known as Auto-Negotiation. Auto-Negotiation requires only 2 pairs of conductors. The maximum available capability is subsequently shared between the devices.

"If the maximum shared speed for both sides is agreed to be 1 G then the link needs 4 pairs of conductors to sustain that speed. If only 2 conductor pairs are available, both devices will conclude the 4 pair connection is unreliable in way, and will then enter a search mode to find a speed that will work on any available 2 pairs of conductors."

The conclusion we reached was that "search mode" is not as fast as wireless due to that negotiation. If the hardware implemented to test that link is determined to be sufficiently compromised, it will abandon that 4 pair operation and revert to a lower speed which is sustained for the duration of the connection.

From that discussion I learned that the apparently simple Ethernet connection is not so simple after all, and yes it confirms our suspicion that your office-to-router link is not conforming to Gigabit Ethernet requirements. It may not even be conforming to 100Base-T requirements. I suspect you will ultimately find one or more broken or intermittent conductors or connections, and that repairing them will solve the problem. The required number of twists per inch is specified. Exceeding even the maximum length of any particular un-twisted pairs of conductors can affect its negotiated speed.

Since that discussion I learned a lot more about Gigabit Ethernet. The paraphrased notes I created at the time are exceedingly brief.

Great! Glad to hear it.

Thank you for the explanation of the reason. When you find an intermittent connection due to brittle conductors it's very likely all them are in need of attention, so the electrician did it right.

I imagine it didn't cost very much.

A reasonably qualified electrician should be able to diagnose the problem in no time. Don't go crazy searching for a network engineer or something.