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The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The wizard being unconscious does not affect anything the familiar could already do on its own as it is perfectly capable of independent action. That means that a wizard could have arranged for the familiar to follow actions in the case of the wizard getting knocked out.

What actions they are capable of taking is going to depend on the situation, the statistics (especially Intelligence score which is usually between 2 and 4), and the physical characteristics of the familiar. Your DM will have to work with you to decide what a familiar can do and what it actually does. For more discussion on this see What can a familiar actually do? for more discussion.

The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The wizard being unconscious does not affect anything the familiar could already do on its own as it is perfectly capable of independent action. That means that a wizard could have arranged for the familiar to follow actions in the case of the wizard getting knocked out.

What actions they are capable of taking is going to depend on the situation, the statistics (especially Intelligence score which is usually between 2 and 4), and the physical characteristics of the familiar. Your DM will have to work with you to decide what a familiar can do and what it actually does. For more discussion on this see What can a familiar actually do?.

The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The wizard being unconscious does not affect anything the familiar could already do on its own however as it is perfectly capable of independent action. However, what things it could do to help the wizard are very dependent on the situation and outside the scope of this question. See What can a familiar actually do? for more discussion.

The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The wizard being unconscious does not affect anything the familiar could already do on its own as it is perfectly capable of independent action. That means that a wizard could have arranged for the familiar to follow actions in the case of the wizard getting knocked out.

What actions they are capable of taking is going to depend on the situation, the statistics (especially Intelligence score which is usually between 2 and 4), and the physical characteristics of the familiar. Your DM will have to work with you to decide what a familiar can do and what it actually does. For more discussion on this see What can a familiar actually do? for more discussion.

The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The unconscious condition says:

An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings.

and incapacitated says:

An incapacitated creature can't take actions or reactions.

So:

1. No, the unconscious wizard can't speak or communicate because they are unconscious. They are literally passed out, unaware of anything and unable to do things. The telepathic bond is still there but one end of the line is dead.
2. No, the wizard is unable to communicate. You cannot instruct a familiar to take certain actions without communicating with it.
3. No, the wizard casts the spell, the familiar only delivers it.

When you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.

(emphasis mine)

Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.

The wizard being unconscious does not affect anything the familiar could already do on its own however as it is perfectly capable of independent action. However, what things it could do to help the wizard are very dependent on the situation and outside the scope of this question. See What can a familiar actually do? for more discussion.