Almost every Pro Micro based keyboard should work with the nice!nano. If you want to use the nice!nano on an existing keyboard, it will be much more complicated and will require "hijacking" the existing matrix.
The nice!nano will NOT work with the Gherkin or Helix unless you leave the RAW pin disconnected from the board.
They don't charge or power each other. Each side has an individual Li-Po battery connected to it via 2 extra pins at the top of the board (called B+ and B-).
The nice!nano has a Li-Po charger built in that uses the USB-C port to charge the Li-Po at a rate of 100mA.
This is highly dependent on the battery size and features of the keyboard. The power profiler from ZMK offers a good estimate of the battery life you can expect.
No, there's a connection via BLE between the two boards. The central half reports back the keystrokes of both the central (usually left) and peripheral (usually right) sides.
This is complicated. Nordic's nRF52 line has some licensing issues with its SDK making it not possible to be upstreamed to the main QMK repo. Instead, we rely on other firmwares like ZMK or BlueMicro that offer a great deal of functionality with full legality and wireless focus.
Joric's nRFmicro wiki is an amazing resource to get some basic and advanced information on the nRF52 line in terms of keyboards: https://github.com/joric/nrfmicro/wiki
The nRFmicro is extremely similar to the nice!nano. The main difference is depending on the version of the nRFMicro, the power system would be slightly different from the nice!nano. From a usability standpoint, very little is different. The nice!nano exposes more pins and is thinner than older versions of the nRFMicro. The biggest difference is that the nice!nano is prebuilt and has a large user base, which therefore has a bigger support community. The BlueMicro is basically the same story except for the nRF52832 versions don't support USB.