Selfhosted

Basically, no:

It can cause some wackiness… basically you will need to maintain that old domain forever and everything will still refer to that old domain.

For example, your post looks like this from an ActivityPub/federation perspective:

{
    [...]
    "id": "https://atosoul.zapto.org/post/24325",
    "attributedTo": "https://atosoul.zapto.org/u/Soullioness",
    [...]
    "content": "<p>I'm curious if I can migrate my instance (a single user) to a different domain? Right now I'm on a free DNS from no-ip but I might get a prettier paid domain name sometime.</p>\n",
}

The post itself has an ID that references your domain, and the the attributedTo points to your user which also references your domain. AFAIK there is no reasonable way to update/change this. IDs are forever.

It would also break all of the subscriptions for an existing instance, as the subscriptions are all set to deliver to that old domain.

IMO your best bet would be to start a new instance on the new domain, update your profile on the old one saying that your user is now @Soullioness@newinstance.whatever and maintain that old server in a read-only manner for as long as you can bear.

You can, but other servers will not recognize it as the same server.

There are however ways to run your server on a subdomain and make it appear as if it's still the original domain. Is that what you're looking for?

Not impossible but difficult. I would recommend against it.