How do protestants respond to apostolic succession?

Protestants would answer that apostolic succession does not mean a denomination has some sort of linkage or appointments from today backward to one of the apostles. Apostolic succession is not some continuous line of Person X to Person W to Person V and so forth back to Person A.

The Catholic Church uses this lineage, like a blood line of yore, to prove its apostolic succession.

The principle underlying the Roman claim is contained in the idea of succession. "To succeed" is to be the successor of, especially to be the heir of, or to occupy an official position just after, ... Apostolic Succession NewAdvent

Instead, Protestants would answer that apostolic succession is the biblical concept that a person is teaching what the apostles taught, instructing the same as apostles. Here is Paul:

And the things that thou hast heard of me among many witnesses, the same commit thou to faithful men, who shall be able to teach others also. 2 Tim 2:2

To take this a step further, very early on, anyone could know that those at Rome who claimed to teach the same, that is, claimed apostolic succession, were off base.

6. But that they who are at Rome do not observe those things in all cases which are handed down from the beginning, and vainly pretend the authority of the apostles;2925 any one may know also from the fact, that concerning the celebration of Easter, and concerning many other sacraments of divine matters, he may see that there are some diversities among them, and that all things are not observed among them alike, which are observed at Jerusalem, just as in very many other provinces also many things are varied because of the difference of the places and names.2926 And yet on this account there is no departure at all from the peace and unity of the Catholic Church, such as Stephen has now dared to make;2927 breaking the peace against you, which his predecessors have always kept with you in mutual love and honour, even herein defaming Peter and Paul the blessed apostles,2928 as if the very men delivered this who in their epistles execrated heretics, and warned us to avoid them. Whence it appears that this tradition is of men which maintains heretics, and asserts that they have baptism, which belongs to the Church alone. Firmillian to Cyprian re Heretical Baptisms

Pope Stephen had departed from "teach the same". Firmillian and anyone else knew this. It had begun within 100 years of Christ's ascension regarding Easter and other sacraments. Rome had departed. As such, Firmillian and others had apostolic succession, not Rome.

Again, as Paul instructed us, saying:

And I thank Christ Jesus our Lord, who hath enabled me, for that he counted me faithful, putting me into the ministry; 1 Tim 1:12

God is the one who puts faithful people into the ministry of grace.

Apostolic succession is simple enough. Teach the same as apostles and you have apostolic succession.

PS A comment suggested that I had changed the definition of "apostolic succession" from the Pauline "teach the same" to some sort of mystical bloodline.

There are numerous problems with the view of a mystical bloodline from bishop to bishop.

The Orthodox Church claims apostolic succession as does the Catholic Church. Obviously, they have departed from the Pauline "teach the same" to simply claim a mystical bloodline of bishop to bishop who may teach anything they want.

Such was the Firmilian and Cyprian claim against the Catholic (and at the time Orthodox) successors. They were not teaching what was handed down. Anyone may know.

Also, Paul tells us what is required of a successor. It wasn't some mystical appointment, but rather faithful men who would teach the same and hand down that teaching to faithful men who teach the same.

Irenaeus understood.

1. Such, then, is their [the heretics] system, which neither the prophets announced, nor the Lord taught, nor the apostles delivered, but of which they boast that beyond all others they have a perfect knowledge. They gather their views from other sources than the Scriptures; AH Book 1 8

Prophets, Lord, apostles to faithful men who would teach the same.

We are warned to watch of our own selves men from ourselves who teach other than what was handed down from the beginning. (emphasis mine)

Then, referring to the evil teachers who should arise, he [Paul] said: “I know that after my departure shall grievous wolves come to you, not sparing the flock. Also of your own selves shall men arise, speaking perverse things, to draw away disciples after them.” AH III XIV

Apostolic succession was and remains "teach the same". To paraphrase Jude's "contend for the faith once delivered", Irenaeus says this.

And undoubtedly the preaching of the Church is true and stedfast,4620 in which one and the same way of salvation is shown throughout the whole world. AH Book V XX

All additions and subtractions from the faith once delivered that happen because of appointed men in history are a departure from apostolic succession of original faithful men.

PPS Let's take a closer look at the OP's quotes to see what is actually being said. As we do, keep in mind that neither Cyprian, Firmillian, or those like Polycarp, Polycrates, Melito, the Orthodox Church, and others gave zero credence to the false idea that "apostolic succession" meant anything like a bishop is valid simply because he can trace a mystical bloodline back to an apostle. What mattered to all of them to have apostolic succession is faithful men who taught the same as apostle from the Bible to today. AS is "teach the same".

Hegesippus

1. Hegesippus in the five books of Memoirs1227 which have come down to us has left a most complete record of his own views. In them he states that on a journey to Rome he met a great many bishops, and that he received the same doctrine from all.

Eusebius states the same definition of AS. Hegesippus met bishops. So? The key fact is they taught the same doctrine as apostles. Again, this follows the Pauline "teach the same". Your pedigree matters not.

For there is no respect of persons with God. Rom 2:11

Let's turn to Clement's letter in which he says this.

It is right and holy therefore, men and brethren, rather to obey God than to follow those who, through pride and sedition, have become the leaders of a detestable emulation. Chapter XIV

This is what Peter said too.

Finally, let's turn again to Irenaeus to see his definition of apostolic succession. It is the same as all the others; that is, teach the same.

1. It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about. AH 3:3:1

Irenaeus does not say, listen to the bishops because they were appointed by other bishops. No, the key is the fact that certain bishops were teaching the same as the apostles. Obviously, any bishop can claim to be a bishop appointed by a bishop blah blah, but apostolic bishops are those who do not veer away, invent doctrine, and add or subtract, rather apostolic bishops are those who teach the same faith once delivered.

As I said before, apostolic succession regardless of your denomination means the same for 2,000 years. Teach the same as apostles and you have apostolic succession.

Obey God. Not a so-called line of appointed men.

The primary scripture that I have seen presented in favor of the idea of apostolic succession - especially with regards Peter and the Papacy - is this passage:

Matthew 16:17-19 Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven. 18 And I tell you that you are Peter,[a] and on this rock I will build my church, and the gates of Hades[b] will not overcome it. 19 I will give you the keys of the kingdom of heaven; whatever you bind on earth will be[c] bound in heaven, and whatever you loose on earth will be[d] loosed in heaven.”

Certainly a powerful passage on its own. However, Peter's own relevant powerful interpretation and application of this idea go ignored!

1 Peter 2:4-8 As you come to him, the living Stone—rejected by humans but chosen by God and precious to him— 5 you also, like living stones, are being built into a spiritual house[a] to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ. 6 For in Scripture it says:

“See, I lay a stone in Zion, a chosen and precious cornerstone, and the one who trusts in him will never be put to shame.”[b]

7 Now to you who believe, this stone is precious. But to those who do not believe,

“The stone the builders rejected has become the cornerstone,”[c]

8 and,

“A stone that causes people to stumble and a rock that makes them fall.”[d]

They stumble because they disobey the message—which is also what they were destined for.

So on the one hand, yes Peter is a stone upon which the church is built. On the other hand, Peter does not think himself the Corner Stone (clearly Christ). His being a stone upon which the church is built is not a unique designation. Rather, he speaks of this as a general attribute of the entire body of Christ - we are all living stones, and we are brought together to compose the church.

If it is a question of the validity of churches/Christians who don't try to draw their authority via some claim of succession back to the early church - I would direct you to this passage:

Mark 9:38-41 “Teacher,” said John, “we saw someone driving out demons in your name and we told him to stop, because he was not one of us.”

39 “Do not stop him,” Jesus said. “For no one who does a miracle in my name can in the next moment say anything bad about me, 40 for whoever is not against us is for us. 41 Truly I tell you, anyone who gives you a cup of water in my name because you belong to the Messiah will certainly not lose their reward.

We see here the disciples being concerned about the validity of someone going forth and doing miracles in Jesus' name. They even told him to stop - this man wasn't one of them and they had not authorized him to do any such thing. They felt like they had some kind of ownership over the ministry.

But Jesus rebuked his disciples and told them not to stop the man from doing good in his name. That man was not an enemy of the ministry - but a fellow minister. Jesus and his ministry were not the property of the disciples back then - and they most certainly aren't the property of those who claimed to succeed the disciples and apostles today.