Oh that looks like fun trying to type anything in right now. Can you see if the situation changes when you plug in a PS/2 keyboard and type only on that?

The keyboard controller circuitry is located underneath the large standby NiMH battery which often leaks and damages things - so it may not be just the keyboard membranes / cables causing problems here. It looks like you may have caught it early enough that perhaps it could be got back to working, but I know Nexxen on here has had a similar problem with his Toshiba 4xxCDS/CDT and I'm not sure that's resolved yet.

There are no leak from the batteries. Now i have disconnected the internal keyboard and connected external one via PS/2 after turning it on, looks like some key is pressed (the same was with internal keyboard) and sound starts beeping... https://youtu.be/10iiezaeca0

That electrolyte can travel far, there can be corrosion on the back of the motherboard, but from your video I can see it does look clean in the sockets so that's less likely. There could be corrosion underneath the standby battery tray, that's where I found some on the 400CDT I fixed that would show a keyboard error on boot. With that one, some of the electrolyte from the standby battery dripped onto the small square chip near the keyboard controller, which I think connects the KBC into the rest of the system: Re: Toshiba Satellite 200CDS 810 - troubleshooting and general info (KBC error - trying to solve, help needed)
Please note that I was calling it an H4044 in error possibly, it's definitely an H4066 (TC74HC4066AFT) when I check a similar board now.

The other day I had the PS/2 keyboard going crazy on a board I fixed, where I had one of the KB-CLK & KB-DATA wires shorting to ground and from bootup it made angry beeps / random key presses.
If you've got a multimeter, you could check whether the CLK or DATA pins connect to either 5v or ground:

On my 400CDT, the CLK and DATA pins both show megaohms of resistance to 5v and ground, so pretty much not connected, but not completely disconnected either.

Thank you for the information! I will try to completely disassemble it and check all the board for corrosion, also will take some pictures.

Just take the LCD + palmrest off first, I think it's more likely there's some electrolyte on that surface since the battery terminals look pretty fresh.

I checked everywhere but found no signs of corrosion anywhere