I think your soldering style here could potentially be a cause of the difficulty.
My suggestion would be:
De-solder all components
Remove excess solder with a desolder pump, or solder wick
If possible, clean away any residue with a spray of isopropyl alcohol and a clean toothbrush (obviously not one you intend to use for dental care!)
After that, it would be very helpful to see a photo of all the components laid out flat, along with your soldering equipment, spare wire, and any other relevant supplies.
This will help us evaluate the state of the project and figure out the best way forward.
I think desoldering and after that maybe soldering everything on again would be too much work since everything is so small. Ill just try the sd reader out with another arduino somehow.
This way it is easy to make changes, until you are confident about your hardware setup. By soldering "header pins" to your Pro Micro, you can easy fit it into a breadboard.
When you are ready to make your project more permanent, rather than tightly wiring the modules together, it is more common to mount it on some kind of "protoboard", or "stripboard"
Unless you have a particularly good reason, tightly wiring everything directly together, like in your photo, can cause a lot of headache. Breadboards and protoboard are common for a very good reason: they make life easier!
I still would strongly encourage you to re-solder the project. From the information you have provided so far, the solder-work is definitely the most promising thing to investigate.
My concern is about small solder bridges, bad joints, or even flux-residue that you might not be able to spot for yourself. I would have liked to see the parts cleaned first, to assess their state. I would have liked to see your solder, soldering iron, and wire, to assess if it was suitable for the task. I would have encouraged you not to reassemble it in the same manner as the photo you shared earlier.
