He could just run el7 based distro as they use the 3.10 kernel and are still supported until 2024.
I wanna thank everyone for the suggestions and help, really - thank you!
In order to update you, I will say that I have tried most of the suggestions (as much as my free time allowed me) - and nothing could completely solve the problem. Yes, I could run distros with lower kernel versions, but since I really eanna run latest Kali/Parot OS versions - that is not solution for me.
In the end I solved th problem by installing Windows 10 and selling the laptop since it is perfect Windows machine with no issues whatsoever.
I will buy something more compatible and lighter to carry around. 15" preferably.
Thank you all once again.
Cheers.
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Seems weirdā¦
I have t540p (practically the same laptop) and it works very well with manjaro (5.9 kernel). I was also running everything from ubuntu 16.04 forward on it without any issues.
Thinkpads have generally very good support for linux, so it seems weird you had that experience.
It seems you managed to not loose too much money or time, so best of luck in your search! 
I donāt know , mateā¦
What is the BIOS version on your 540p?
Thank You
I think below is all that could be relevant
UEFI: GMET89WW (2.37) [2019-08-23] (uefi secure boot disabled, intel smart connect disabled)
Embedded cntrl version GMHT30WW (1.15)
ME FW: 9.1.45.3000
I usually update bios when reinstalling any OS (usually Windows dual boot, like once ~2yearā¦) just in case.
Since ~2016 iām using this laptop, and i donāt recall any stability issues.
Except nVidia being PITA under linux, and touchpad driver being weird (can not disable touchpad without disabling ābuttonsā for trackpointā¦), i have no other complaintsā¦
My 440p used to work perfectly with linux and 2.37 BIOS, problems with linux started after I updated BIOS to a higher version - so my advice to you, no matter what you do, do not update that BIOS, unless you are ready to only use linux distros up to 4.17 kernel I have tried to downgrade BIOS, but Lenovo put restriction to that, soā¦ dont touch that BIOS
Regarding th touchpad, I never liked touchpads on that 44x and 54x thinkpads, so I replaced it with one from the T450 - much better and with no issues.
Huh, thanks for the info. I thought my bios was fairly new, didnāt even check before replying aboveā¦ My badā¦
This is now more or less my ābackupā work computer, so i have a sudden itch to update itā¦ Might do it in December when iāll have more time, just to double check the theory and complain to lenovo to release 2.39ā¦
Will let you know when i do itā¦
The latest BIOS on my T440p was 2.55, I had hopes that one will fix the problem, but noā¦
I would have suggested downgrading the bios firmware until the problem went away. But guess it donāt matter now. Really you shouldnāt update bios unless there is a specific need to such as supporting new hardware you intend to get or fixes for a problem you are having.
I tried to downgrade, however Lenovo disabled that posibility after certain BIOS version. According to Lenovo, it is out of the security reasons I could probably go around thatm but just did not have time to deal with that and did not wanted to risk to brick my laptop, so decided to sell it.
An obvious thing to try is clearing your BIOS/UEFI settings (reset to defaults) after updating to resolve any incompatibilities between new firmware and settings/efivars/string tables/etc. made with the older version.
Hey.
I have been facing the same exact problem with Linux, on Bios 2.56, but I also have some difficulties with windows 10, too. Whenever I boot into them, after the initial windows logo the screen goes black and remains that way unless I change my screenās brightness, and then it works normally.
On Linux, I can run ubuntu 18.04.6 with an early 4 something kernel,but with anything newer the installation media either will outright not work, or it will not boot past the grub after installing.
If this is because of the bios, since I can not downgrade, do I have to buy a new mobo or a cpu?