MDIO communication happens in my test only when the Interface is configured at start-up for a short while. So most of the time it is silent.
The configuration of Ref_Clk pin is done in eth.c and not related to eiterh LAN8720 or LAN8742. So oen has to analyze the source code with the STM32F743 reference manual at hand and determine, where ref_clk is configured.

I do not know which code you used for the RPi Pico as logic analyzer, but this one looks ok: https://www.hackster.io/markkomus/using-a-raspberry-pi-pico-as-a-logic-analyzer-with-pulseview-e12543#code
The sample rate is 1MHz. You may be able tweak it up, but at that rate you see at least some transitions, if there is communication at all.

Edit: Or this one: https://hackaday.com/2022/03/02/need-a-logic-analyzer-use-your-pico/

I fixed it. Check pull request #13422. @robert-hh please stay tuned, I am going to check Internet via LAN, and thanks for your replies.

I just tried this Logic Analyzer Software for RPi Pico, and it worked "out-of-the-box". https://github.com/dotcypress/ula/blob/main/readme.md

Great. So it was the missing REF_CLK. I had that once, and I searched for a while. As soon as I knew it was the problem, I was able to find the fix.

It would be interesting if eth_process_frame() and eth_netif_output() are ever called. The first one is called from the receive interrupt, the second afgte processing a n input frame by LWIP and sending a response. There is an internal trace function, which is enabled by calling eth_set_trace(), and that is enabled with nic.config(trace = flags), with flags being a combination of the bits:

#define TRACE_ASYNC_EV (0x0001)
#define TRACE_ETH_TX (0x0002)
#define TRACE_ETH_RX (0x0004)
#define TRACE_ETH_FULL (0x0008)

b.t.w. The settings for gateway, netmask and DSN are the default values, set in 👍

STATIC void eth_lwip_init(eth_t *self) {
    ip_addr_t ipconfig[4];
    IP4_ADDR(&ipconfig[0], 0, 0, 0, 0);
    IP4_ADDR(&ipconfig[2], 192, 168, 0, 1);
    IP4_ADDR(&ipconfig[1], 255, 255, 255, 0);
    IP4_ADDR(&ipconfig[3], 8, 8, 8, 8);

@robert-hh sorry if bothering you, I did not seem to find how to trace an internal function, so do you know how to do it? I am beginner to this, so please help me, and sorry again if I bother you. Here's piece of code that I tried:

import network
nic = network.LAN()
nic.config(trace=8)
nic.active(True)

try:
	nic.ifconfig("dhcp")
except Exception as e:
	import sys
	import io
	buf = io.StringIO()
	sys.print_exception(e, buf)
	print(buf.getvalue())

Setting e.g. nic.config(trace=15) enables the trace functions already implemented in the code. See function eth_netif_output() or eth_process_frame(), which call eth_trace(). which calls netutils_ethernet_trace() to print the frame content. Printing the inpit may fail, since it is called from within a IRQ handler.

So netutils_ethernet_trace() by default calls MP_PYTHON_PRINTER (the stdout). I just don't know how to read stdout. Sorry if this is very easy answer, I didn't seem to find it.

I did not seem to find anything to help. Only current difference from quick look at STM article for using ethernet, all pin freqs are set to very high, but that also did not seem to help. I will try go deep into that STM article and code. @robert-hh here is some reference issues similar to mine, and my code:

1. Discussion (that guy have same problem, only LAN chip is different): Using PHY module DP83848 with Micropython #10141
2. Issue (at me resloved): knock-off LAN8270 board on STM32F407VET6 black board #6795
3. The article (it is H7 series, but worth looking): https://community.st.com/t5/stm32-mcus/how-to-create-project-for-stm32h7-with-ethernet-and-lwip-stack/ta-p/49308
4. My code: https://github.com/ukicomputers/Fusion_STM32

Also, what lan.status() values each mean?
I also noticed lwip MPY libary on my board, if anything means it for troubleshooting.

The possibly undefined state of 7,8,11 seems not critical, since the mode is set by code.

It seems that this is not true, because these are set in register 4 of the LAN8720, which is not touched by the code.

Besides that, I hardly get any further by staring at the code, and lacking an appropriate hardware I cannot test. The only board w/o Ethernet attached for purchase seems to be the STM32F429 DISC.

The F42/F43 series are very similar, so probably you will expect if you try (I cannot thank you so much, thanks again @robert-hh) same problem as I do. And the referenced issue upfront, is for F40 series, so Fxxxx series MCU's probably got same problem for LAN8720. Thanks!

I will also try out this, maybe STM's ethernetif.c/h will solve the issue at initializing: https://github.com/STMicroelectronics/STM32CubeF4/tree/master/Projects/STM324xG_EVAL/Applications/LwIP/LwIP_HTTP_Server_Raw

Also, I did not seem to capture any traffic when LAN is initialized. When pinging, immediately when I start ping, the RX light starts blinking, but I get Destination Host Unreachable, I think because of unable of replying. I searched with Wireshark.

That sounds really excellent. So it was in the end the missing REFCLK.
I wonder why you had to change the clock config sections. If the basic MicroPython firmware works with the F437, isn't it sufficient for the clock to use the MicroPython code:

import machine
machine.freq(150_000_000)

With the MCO01 setting you should get the 50MHz clock. Note that you can use machine.mem32() and machine.Pin() to configure MCO01. No strict need to change the driver. The code would look like:

from machine import Pin, mem32, freq
  # set the PLL freq to 150 MHz
freq(150_000_000)
# configure Pin A8 to Alt mode PP
p=Pin(Pin.cpu.A8, mode=Pin.AF_PP, alt=0)
# Enable PLL clock / 3 at Pin A8
mem32[0x40023808] = 3 << 21 | 5 << 24

Ok, the web interface and communication is working now without any problems (I am using web server on GH: @troublegum/micropyserver, and it works perfectly). Thanks for everything @robert-hh. If there were any problem, I will write here. Thanks again.

Also, I do not seem to get urequests and other similar libraries even if I enabled LWIP, MBEDTLS... Do you know what may be the problem? btw, this is link of my repo: https://github.com/ukicomputers/Fusion_STM32

urequests is a python library in https://github.com/micropython/micropython-lib/tree/master/micropython/urequests. You can include it into the firmware as frozen bytecode by adding the line:
require("bundle-networking")
to the board's manifest.py. See a sample in the board files for PYBD_SF2.
Edit: See also: https://github.com/micropython/micropython-lib/blob/master/micropython/bundles/README.md

Yeah, I forgot :), I changed everything + .mk file, but I forgot it somehow. Thanks again for everything!