Barring telepathy or direct observation, what information does Find Familiar provide the caster regarding the familiar's state?

Question

This question has its genesis in a common situation: a PC sends his familiar reconnoitering.

The familiar travels beyond the range of both direct observation (vision, hearing, etc.) and find familiar's telepathic communication (see PHB p. 240). When the familiar does not return at the appointed time, the PC comes to suspect something's wrong. And in point of fact, the familiar has been killed off-camera. The player asks how his PC, who has no other relevant spells and no magic items, might determine whether the familiar is alive or dead.

Debate ensues, leading to a deeper question about the nature of find familiar. Once past the limits of telepathy or direct observation, does a familiar turn into a black box? In other words:

What, if anything, can the caster of find familiar ascertain about the state of his familiar purely by dint of the spell itself?

For example:

• Does the caster automatically know whether the familiar is alive or dead?
• Does the caster automatically know whether the familiar is in its pocket dimension?
• If the caster uses an action to dismiss the familiar into its pocket dimension, does the caster know that the dismissal succeeded?
• If the caster attempts to use an action to dismiss the familiar into its pocket dimension, but the familiar has been killed and has "disappear[ed]," what does the caster learn? That the familiar is dead? That the familiar has disappeared, but not specifically that it's dead? That the dismissal failed? Nothing?
• What if the familiar is already in its pocket dimension? Does the caster learn that the familiar is in its pocket dimension? That the dismissal failed? Nothing?

Answer 1

You can dismiss, then re-summon the familiar (which will fail if the familiar is dead)

Nothing in the description of the spell says that the caster knows the state or location of the familiar, except that some options will only work if the familiar is within 100 feet. The general rule is that spells do what they say, and no more. That means that you have no special information on the status or location of your familiar.

There is no official rule on if you know that your dismissal actions succeed, so that would be a DM call; however, the safe assumption is that since it doesn't specify anywhere that you do know that the action failed, you don't gain that information.

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What you can do, however, is (temporarily) dismiss the familiar. There is no distance (or planar) limit on this, so it will always succeed (if the familiar is still alive). You can then use a second action to make it re-appear within 30ft of you. If the familiar was dropped to 0 hp (or killed), the actions will fail and you know that your familiar is gone.

Re-casting find familiar by itself won't help here, since casting the spell while the familiar is alive will only cause it to gain a new form.

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There are two types of "dead" for a familiar

As the spell describes, a familiar that is brought to 0 HP isn't killed as such; instead it simply disappears. Casting find familiar again will cause the same familiar to re-appear. This will be the most common way for familiars to be "killed', but it is a rather temporary affair.

As this question's answers teach us, being reduced to 0 hit points isn't the same as being killed. The familiar might instead have been hit with a power word: kill and be actually killed off. The spell doesn't specify what happens in this case, but since it is not the same as being dropped to 0 hit points, the general rule for creatures being dead would apply, and find familiar can't bring it back.