I’m in the middle of moving house and have my priorities sorted (I upgraded to Papyrus Author+), but we are going to be without internet for a couple of weeks.
Will the software work ok without being connected?
Congrats–Here’s wishing you a hassle-free move! 
Papyrus Author runs without an active internet connection, but it needs an occasional online check-in for authentication and updates.
If you’re away from your internet connection for a longer period, you can briefly share your phone’s connection if needed. The check-in takes just a few seconds, and after that it works offline for a good while again.
Does your cell phone and plan support tethering? I often use this and a cloud service to move my current project between my laptop and my desktop if I forget or can’t move it manually before work.
As for the root of your question, yes. It will not bother you for about two weeks in offline mode; but after 14 days (I think) it’ll start saying nope! Gotta check you’ve paid up!
I think it does this so people can’t pay the sub once and then gleefully cancel their sub and keep writing forever just by isolating their device from the internet. xD
How long is a good long while? I will be using the free version, on a computer with no internet capability whatsoever. Do I still have to log in occasionally?
Hi, welcome to the Community! 
Unfortunately, Papyrus Author won’t work on a computer that doesn’t have an internet connection, as you need to log in with your account after installing it, and for that moment Papyrus Author needs to check up on the internet.
If the computer can be connected to a phone’s internet for a few seconds, you can log in using the shared hotspot connection.
I Hope you’ll find a way to work with Papyrus Author! 
What about installing Papyrus on a USB memory stick on a computer with internet connection and making it run from there? I think there’s even a setting to keep all preferences in the program folder. Then the stick can be used on the computer without the internet connection and it should work for quite a while.
Edit: The option to save all settings in the program folder is here:
Preferences > User Interface > Miscellaneous > Store all settings in Papyrus folder
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