"Do unmanaged switches still have a software layer to operate?"
Maybe, maybe not. If they are running software, likely burnt into some kind of ROM.
"How an unmanaged switch operate in case of network congestion? They still have a buffer?"
Would expect them to buffer some frames, and when buffer filled, drop frames. Buffer resources likely less than on managed switch mostly because unmanaged switches tend to provide less expensive switches.
"Where can I find more information on how the switches work (not necessarily unmanaged), especially the software part?"
Depends on what aspect of their operation you're interested in. Different switches likely take slightly different approaches solving the same issues. As to software, they would often be considered proprietary, so you're not going to find out much except possibly except reviewing generic "solutions" for the common "software issues". For example, supporting STP is likely done via software. (BTW, regarding software, also keep in mind the underlying hardware, supporting software, likely also varies much from switch to switch.)
