Well, for Edge we are currently trying to get the extension into the official edge store (since the chrome extensions seems to be transferable) and in this process are also checking how our customers can roll it out within the company. I hope I can give you an update next week.
Regarding distribution itself - check out "Distributing Extensions" in MS docs here: https://docs.microsoft.com/en-us/microsoft-edge/extensions/extensions-for-enterprise
Hint: (no official support!): by either using sideloading (and zipping the local chrome extension on your disk) or by installing from the chrome store you could already use the chrome extension in edge too for a single browser.
Hey, the extension is the same, therefore not necessarily.
Should you want to roll it out company wide and as of now I can only offer you this: https://docs.microsoft.com/en-us/microsoft-edge/extensions-chromium/developer-guide/alternate-distri...
We already plan to try a company wide roll-out ourselves in the upcoming weeks, which we haven't because we first needed it to be published to the official store (which it now finally is).