What happens (in detail) when a thread makes a system call by raising interrupt 80? What work does Linux do to the thread's stack and other state? What changes are done to the processor to put it into kernel mode? After running the interrupt handler, how is control restored back to the calling process?
What if the system call can't be completed quickly: e.g. a read from disk. How does the interrupt handler relinquish control so that the processor can do other stuff while data is being loaded and how does it then obtain control again?
Good questions! (Interview questions?)
The int $80 operation is vaguely like a function call. The CPU "takes a trap" and restarts at a known address in kernel mode, typically with a different MMU mode as well. The kernel will save many of the registers, though it doesn't have to save the registers that a program would not expect an ordinary function call to save.
Typically an OS will save registers that the ABI promises not to change during procedure calls. The stack will stay the same; the kernel will run on a per-thread kernel stack rather than the per-thread user stack. Naturally some state will change, otherwise there would be no reason to do the system call.
This is usually entirely automatic. The CPU has, generically, a software-interrupt instruction that is a bit like a functional-call operation. It will cause the switch to kernel mode under controlled conditions. Typically, the CPU will change some sort of PSW protection bit, save the old PSW and PC, start at a well-known trap vector address, and may also switch to a different memory management protection and mapping arrangement.
There will be some sort of "return from interrupt" or "return from trap" instruction, typically, that will act a bit like a complicated function-return instruction. Some RISC processors did very little automatically and required specific code to do the return and some CISC processors like x86 have (never-really-used) instructions that would execute dozens of operations documented in pages of architecture-manual pseudo-code for capability adjustments.
The kernel itself is threaded much like a threaded user program is. It just switches stacks (threads) and works on someone else's process for a while.