I fixed most of the issues. Starting with analog sticks, I had to desolder the connectors to cut GND trace, solder the connector back, then cut VCC and wire it the other way around. And it's working, but as I thought L3 and R3 buttons weren't working anymore, because they were conected in the analog sticks to VCC instead of GND.
I was thinking how to fix the buttons, and found that hex inverter might fix this problem. So I bought one the next day, wired it up, and now the buttons were always pressed, but pull-down resistor on the output of the inverter solved this issue, and the buttons are working again.
Later I soldered second stick to the inverter, and moved it to a different place, but I forgot to take a photo.
As for the LCD buttons, I discovered that they were working all the time, but I soldered the resistors in the wrong places, so i soldered them where they should be and it's fine now.
After that last problem was the SD card slot. I spent few hours on it, I tried everything and the best I can get is that the console sometimes shows what's on the SD card, but i can't run anything from it. But it was working before the assembly when I connected USB slot to it and ran the padtest off of the pendrive.
I think that the issue is either interference (I twisted usb data wires, didn't help) or something wrong with the adapter I soldered. I give up for now, I'll comeback to it in the future when i get the proper IC.
So for now it's the only thing that's not working