R1+ is waterproof up to 1M and the R1 is rainproof. The R1+ also has additional firmware features that the R1 does not have, for example, on the R1+, you can set the GoPro SSID and password.
The R1/R1+ has a built in LCD screen. The P1 uses the GoPro LCD BacPac, which is sold separately.
Yes, you can use the R1/R1+ while it is being charged from an external power source.
Live streaming is controlled by the GoPro camera. GoPro does not support live streaming while recording for some of the high resolutions/frame rates, for example, 4K at 30FPS. The R1/R1+ support all live streaming that the GoPro camera supports and the R1/R1+ firmware will be updated if the GoPro camera starts to support new resolutions/frame rates.
If your device is not recognized by computer, try using different USB cable, or connect to different USB port. If firmware is not updated even though files are copied, make sure follow the instruction in this page https://removu.com/firmware-mac/firmware-r1p/
Unfortunately the GoPro Hero 5 Session and Hero 7 are not compatible with the R1+. Even if some user managed to connect to it, however it is very unstable/unusable due to hardware incompatibility.