It’s because you have a three-phase supply. Each of the three voltage waves are displaced by 16.667 ms relative to each other. To measure the power, your emonTx/emonPi samples each voltage and current wave about 2500 times per second, multiplies each pair of samples together to get the power at that instant, then averages all those readings of power. If you’re multiplying the current by the wrong voltage, you get the wrong answer.

It’s to provide that mains voltage sample, and you will need one, whether you have the emonPi or emonTx. But the emonTx does not need the separate d.c. power supply.

Yes to the first, no to the second. What I should have mentioned before was that by making a wired serial data connection between the emonTx and a Raspberry Pi, you would not need the RFM69Pi adapter that makes the RPi become an emonBase.
You might not need the programmer if you have that serial connection. There’s new software out (I’ve not tried it yet, so I can’t say how straightforward or otherwise it is). The emonBase/emonPi is controlled and set up via a web browser.

Here it is.You will need a separate 5 V d.c. power supply as well as the a.c. adapter with that, due to the additional power needed for Wi-Fi.

Where are you going to run emonCMS - assuming that’s what you want to use to record and view your data? You need a web server that’s continuously available to receive the data as it comes in.

I can’t think which diagram that would be, but no, both need a power source - the interconnection is normally by radio (the RFM69Pi adapter actually receives and turns the radio packet of data into a serial stream for the RPi). Normally, the emonTx is powered from the a.c. adapter, whilst the emonBase needs a 5 V USB power supply. If you do have a serial connection, while both can share a power supply, you still need the a.c. adapter to measure the voltage. The a.c. adapter cannot power the RPi.