Power Automate Desktop absolutely requires internet connectivity. Your flows will not work if you are not connected to internet. That's to answer the last question in the thread.
As to logging in to an account, you can login using a work/school account, in which case you will need a Premium license or a Trial. If you want to use the free version of PAD, tou should login with a free (personal) Microsoft account.
-------------------------------------------------------------------------
If I have answered your question, please mark it as the preferred solution. If you like my response, please give it a Thumbs Up.
I also provide paid consultancy and development services using Power Automate. If you're interested, DM me and we can discuss it.