SD card works OK, here are interesting things that I've learned:

• There are two protocols to communicate with SD card - SPI (slower, but easier) and SD bus, some cards will not work with SPI
• The SD card can draw up to 100mA when writing data!
• As usual, it's good to add small resistors in series to all pins that are accessible from external, user can insert or remove card, so all data ports can be considered to be accessible from external. This is to prevent ESD damages. For SD card those resistors are in range 40-50R.
• It's good to add pull-up resistors to all data lines, otherwise some cards will not work. There are chips on the market with pull-ups and in-series resistors.

Info from: [1], [2]

The prototype is now a bit bigger than before, because I've decided to cannibalize data logger board for Arduino - back when a shipment from China was for free (in Europe) I'v bought a couple of them - just in case. I don't really need therm since I don't do a lot of Arduino, and this saves me on buying SD card slot and RTC clock. I will add an RTC clock to the design. 

The SD slot was a bit problematic, finally I've de-soldered it to see bottom side and how the card is really connected. It's weird for me because they didn't ignore pin #9, it's accessible even when AFAIK it's unused. So the pinout is like this 9, 1, 2, ..., 8, extra pins for check write protection. 

In the next step I plan to setup ESP32, it will involve probably soldering new breadboard to the prototype :)