I get only 4.4 Volts. Could that be too low ?
Can I conclude that there is something wrong with the board ?
I have no idea what you are doing with the Tx Rx stuff, maybe because my Ethernet is built-in, but I have had problems with power lately.
You make no mention of the power supply but I suspect it is via the USB cable, and it could be all down to that. I started with a USB cable, whic was actually OK, but I then tried a converted phone charger, good for 5.3v @ 1amp. It looked good and then the lcd went out. This happened when I started to use the Ethernet. In a separate exercise, I was checking the temperature under the board and it went up by a few degrees when, you'll never guess, yes, when I used the Ethernet.
In short, in the first instance, I was always a bit close to the edge and the Ethernet took that much more power to push me over the edge. The problem was fixed with a proper 9v supply. You may get by using a different USB port. The second instance just demonstrates that the Ethernet makes the regulators work that much harder i.e. a significant increase in power.
Further to this, I have recently changed the motherboard in the desktop. Even without the Ethernet shield, the Mega could not even run the RTC properly. To add insult to injury, the desktop refused to start with the Arduino connected. I transferred the cable to a separate port and the PC will now start OK, but I have not used the Ethernet with that.