Familiarity is not a very precisely defined rules term, so interpretation of it really falls back to precedent, interpretation, and DM rulings. In terms of precedent, the most precise definition of familiarity comes from the Teleport spell:
“Very familiar” is a place you have been very often, a place you have carefully studied, or a place you can see when you cast the spell.
Assuming that "Familiar" is similar but less restrictive than "Very Familiar", then we have some suggestions for how to determine familiarity. As a DM I would rule that without the knowledge "Person A and Person B are the same Person" then "met Person A often" or "can see Person A as you cast" wouldn't be sufficient to say you're familiar with Person B and sending would fail.
I don't think other interpretations are necessarily invalid, but the idea of using Sending with someone really specific as the recipient "Just in case I've met them before and just didn't realize it" seems very against the spirit of the spell and also a huge DM headache.
The other method, "studied carefully", could indicate a method to become familiar through indirect learning, but Teleport helpfully provides other "levels" of familiarity, such as:
“Description” is a place whose location and appearance you know through someone else’s description, perhaps from a map.
which is the lowest level of familiarity, only one step above trying to teleport to a place that doesn't exist. The fact that it's on a chart as a level of familiarity at all could mean it qualifies for "familiar", but I would say Teleport's precedent indicates that you can't cast Sending from just a description.
As for what would happen if someone did manage to Sending them-self, in general the caster of a spell is only an invalid target if the spell's description directly prevents it, either with a special exception (e.g. "a creature other than the caster") or by general rule that includes the caster (e.g. "an enemy"). Sending only provides the "Familiar" restriction, so if the Caster is familiar with themselves (as discussed in the first paragraph) they should be a valid option to receive the message. As for what the result would be, the SRD entry for Sending states:
The creature hears the message in its mind, recognizes you as the sender if it knows you, and can answer in a like manner immediately.
So I think it's clear that the caster would indeed hear the message again in their mind, and they would be entirely certain that it was due to that Sending being sent to themselves (and would then have the option to give a response, though I doubt there would be a reason to). The spell description does not provide any evidence that a failed Sending would be in any way similar to a successful Sending to yourself, nor does it provide any means by which insufficient familiarity would result in the message being sent to the wrong creature, so I think it would be pretty unambiguous that the caster and the intended recipient are the same person.
The Sending to a missing person should fail due to unfamiliarity. How much the caster should know about why it failed is somewhat ambiguous so a DM can rule either way, but the most common interpretation seems to be that the caster wouldn't be able to tell the difference between a Sending that succeeded but received no reply, a Sending that failed due to unfamiliarity, and a Sending that failed due to the 5% failure in cross-plane casts.
I'm fairly confident in my understanding of Sending, though a DM is allowed to change that. For example, if the later dramatic reveal is REALLY IMPORTANT then just deciding that Sending cannot be sent to the caster seems reasonable, as long as that's applied consistently and you don't just make up new reasons out of nowhere why other people can't do the sending or whatever. On the other hand, my definition of Familiar is an interpretation based on precedent and it's reasonable to rule otherwise if you do want this to be possible.