It appears to be 25% more in weight compared to the original battery (~339 grams). So that's only 83 grams. Well within the payload capacity. If it ads 10 extra minutes to the maximum flight time that's about 20% and add a bit more for the...
No, DH is an app on its own. To download and flash the FW is free and you don't need to pay. Please note that the 4G Transmission will not work if you are using any FCC hack. Download the app from below link.
https://drone-hacks.com/
Do you need to use the modded DT DJI FLY app in the USA to use the Mavic 4 Pro with the Enhanced transmission?
I'm going to guess not because it's already in FCC here but I don't know for sure yet