Hello again.
I really doubt anything is wrong with your network card, so I wouldn't take the notebook to a service center. If you have another notebook you could try plugging it in the same way just to verify.
Here's why I think it's fine, but first let me make sure I am understanding this correctly. The navini modem is for a wide range wireless network right? Your thinkpad is plugged directly into it (no hub in between) with an ethernet cable. As a result nearly everyone connected on that network is connected wirelessly. I think what you are seeing is actually poor wireless reception.
Here's an example from the log:
Pinging 5.5.5.1 with 32 bytes of data:
Reply from 5.5.5.1: bytes=32 time=109ms TTL=109
Request timed out.
Request timed out.
Reply from 5.5.5.1: bytes=32 time=109ms TTL=109
4 pings were sent, but only 2 were returned. I think what happened here is two of the pings were dropped or unreadable because of poor reception or even a faulty tower somewhere. For each one there are two opportunities for it to be lost; on the way there and on the way back (the reply).
Here's another example:
Reply from 196.29.212.2: bytes=32 time=1262ms TTL=1262
Reply from 196.29.212.2: bytes=32 time=1119ms TTL=1119
Reply from 196.29.212.2: bytes=32 time=2260ms TTL=2260
Reply from 196.29.212.2: bytes=32 time=1379ms TTL=1379
Ping statistics for 196.29.212.2:
Packets: Sent = 4, Received = 4, Lost = 0 (0% loss)
Approximate round trip times in milli-seconds:
Minimum = 1119ms, Maximum = 2260ms, Average = 1505ms
196.29.212.2 (PASSED)
In this one a reply was received each time, but it took an average of 1505ms. That's 1.5 seconds. This is probably why your internet is slow. In fact, I would guess that 10Mbps is probably faster than what you can receive with the modem anyway.
BTW, just for comparison I pinged that same site and got an average of 106ms.
So basically I think your notebook is fine. I think the poor speed and the test failure are most likely due to the signal the modem is receiving/sending. You could try a different modem, or you might just want to contact your ISP and see if they can help first.