So I bought the DIY kit from AliExpress. (All of the info links about 
the thing will be at the end)

Soldered it (except the LEDs, I want to get the MCU issue resolved 
first). And additionally soldered a reset button in.

It seems that at the moment my MCU (STC12C5A60S2) contains wrong code.
I tried to download the Cube code into it via 4pin UART, but the STC-ISP 
just stuck at "Checking target MCU..."
I checked MCU with oscillograph and it seems to be working fine 
including reset button. It recieves signals on the TxD leg, but does not 
react to them.

Anything that I found so far was that my UART may be 4.25V and changing 
the UART programmer will help, but:
I thought if I connect the board to a power bank and UART at the same 
time then any signal from PC should be solid 5V. Correct me if I am 
wrong on that.

Any help would be appreciated.

Links and info:

Everything LED Cube related (schematic, datasheets, tools, code etc.): 
https://github.com/tomazas/ledcube8x8x8

AliExpress link:
https://s.click.aliexpress.com/e/LlrCw18