Your instance will still exist, and federation should continue as normal if you manage to reclaim the original domain.
If you have to switch to a new one, however, federation will be very awkward. Other instances will essentially treat you as a brand-new instance, and mirrors of old content will be "orphaned" and no longer sync.
When your domain is close to running out, you should either get an email from your registrar asking you to renew, or a payment notification telling you that your domain will be renewed for whatever price automatically.
If the payment fails, the domain will be temporarily suspended. There is a grace period where nobody can buy that domain, allowing you to settle the missed payment. If you do not settle the payment, the domain will be put back up for sale
None of this affects whatever services you're running on your Pi, people just won't be able to connect to it if your domain is suspended.
I'd suggest looking into SSL certificates (Letsencrypt is free) as well as Cloudflare for masking your Pi (your home) IP address from users of your instance - do note this has privacy implications: cloudflare becomes a MITM for your site
Freenom is being sued by Meta (Facebook) at the moment for supposedly not dealing with spam domains. I would not recommend using a Freenom domain if/when they reopen registrations: FMHY had their old Freenom lemmy instance domain seized by Mali's government
I don't think you can change your lemmy instance's domain yet. Afaik there is no official way to do it. FMHY lost their domain (they are using a free domain and lost it) and was attempting to switch to a new domain for their instance and developing a tool to migrate to a new domain, but somehow decided to start fresh and discard their old data instead. No idea what happen with the migration tool they were working on (is it actually working? did they actually released the code?), so save yourself some headache and make sure to never lost your domain, which means don't use free domain because that domain isn't actually yours and can be yanked without any notice.
Yeah, AFAIK ActivityPub itself heavily relies on the domain being part of your identity - so its not really possible to change the domain on any of them, along with other federation implementations such as Matrix.
This is why while Mastodon allows for profile transfers, it doesn't transfer your post content - it simply just sends a signal to your followers to unfollow your old account and follow your new one. The actual content itself is intrinsically tied to your identity on the old domain.
That seems like an oversight. ActivityPub should rely on some sort of certificate or cryptographic signature instead of a domain which might have to occasionally change.
Nothing aside from losing any traffic if people don't know the IP address directly to the server. All a domain does is redirect traffic to the website with an easy to remember name.
The domain is pretty important to Lemmy. If you lose control of it, your instance is effectively dead since the federation will not recognize your traffic until you get the domain back. There's no way to change the domain of an instance so you'd have to start from scratch.