I got it working finally. Now it looks OK. The way I did it is quite simple. I work under Mac OS. Test sequence described in the Wiki looks like this:
1) run LimeSuiteGUI,
2) connect to the board,
3) open self_test.ini,
4) open SXT, press Calculate+Tune,
5) open CLKGEN, press Calculate+Tune,
6) open Modules -> Board Controls -> tick RF loopback ch.A -> RF loopback ch.B
7) open Modules -> FPGA controls -> tick MIMO -> tick WCDMA (load waveform)
8) open Modules -> FFT viewer -> Data Reading -> Packets MIMO, Graphs -> Display Channel -> A&B, START.
That would produce the output seen in the first post.
What fixed it was adding step 4a) open SXR, press Calculate+Tune.
Seems like without doing so limeSDR was out of tune on either Rx or Tx path. That what produced the odd result seen in the beginning of the thread.
Now I am very happy to know my SDR board works fine and I can gently move to learning how to run various applications on it and maybe even write SW programs.