Does a MacBook Air work with an external GPU (eGPU)?

No, not really. Intel MacBook Airs with Thunderbolt 3 lost official eGPU support after macOS Big Sur, and workarounds are buggy with mediocre performance due to CPU and bus bottlenecks. Apple Silicon MacBook Airs have zero eGPU support. Save your money—get a MacBook Pro or a desktop instead.