The hardware level cross-over between logic and physics is of exceptional interest to me, and one of the drivers of my personal investigation of computing more generally.

It is my current understanding of FPGAs that they allow for reconfigurable logical circuits, but still implement those functions using fixed spatial circuits, primarily by dispatching signals down routing traces as specified by LUTs. These spatial circuits, in particular the routing paths, as the physical underpinnings of an FPGA are non-reconfigurable, and form the proprietary core of the product. I have three question, if anyone could provide insight:

(1) Is the above notion correct, or does the act of burning in a bitstream to an FPGA actually physically reroute traces within and between blocks of logic gates?

(2) The latter option in (1) would presumably still require some sort of supporting hardware, itself non-reconfigurable, and subject to proprietariness correct?

(3) Are there reconfigurable devices that offer complete, self-hosted reconfigurability? By which I mean that an unconfigured device must first be bootstrapped (via an external tool) to contain the bitstream burning auxiliary circuitry, but this framework remains no more fixed than the rest of the device. Furthermore, the bootstrapped circuits can update themselves, by creating second-generation support circuitry elsewhere on the device, which access and alters the first generation configuration.

Thank you in advance.

Edit: (4) Given the answers to (1), are there any devices that have reconfigurable routing traces?