Is Everything a Hardware Interrupt?
In general, no, on modern platforms, not everything is a hardware interrupt -- though you could imagine a platform where it is.

On some platforms, hardware interrupts are used for all input events (including, yes, pressing a key on a keyboard or moving a mouse).

However, interrupts come with some performance implications.  Therefore, in modern platforms, we often use a combination of interrupts and other I/O mechanisms.  We might use interrupts for some input devices, but other input devices might use memory-mapped I/O.  For instance, your Ethernet card might use memory-mapped I/O, to write incoming packets into a region of memory dedicated for this use.  Memory-mapped I/O can be combined with interrupts or with polling.

You can find more on interrupts, polling, memory-mapped I/O, DMA, and related topics in a good operating systems textbook.