Must Christians obey the emperor, and pray for him?

Clement of Rome fl. c. 96 ADFirst Letter to the Corinthians 61.1

You, Master, have given them the authority of kingship through your majestic and indescribable might, so that we, knowing the glory and honor you have given them, may be subject to them, in nothing resisting your will. Give them, Lord, health, peace, concord and stability, so that they may administer without offense the government you have given them.

A prayer for the emperors from the Roman church, written under Domitian, who persecuted it.

Justin Martyr c. 100–c. 165 ADFirst Apology 17.1–3

Taxes and contributions we try to pay to your appointed officers everywhere before all others, as we were taught by him. ... "Give to Caesar what is Caesar's and to God what is God's." So we worship God alone, but in other things we gladly serve you, acknowledging you as kings and rulers of men, and praying that with royal power you may also be found to have sound judgment.

Addressed to Emperor Antoninus Pius.

Theophilus of Antioch fl. c. 180 ADTo Autolycus 1.11

So I will rather honor the king, not by worshiping him, but by praying for him. I worship God, the true and real God, knowing that the king was made by him. You will say: Why do you not worship the king? Because he was made not to be worshiped but to be honored with lawful honor.

The line between honor and worship was the line martyrs died on.

Tertullian c. 155–c. 220 ADApology 30.4

We, the Christians, look upward with hands outstretched, because they are innocent, with head bare, because we are not ashamed, and lastly without a prompter, because we pray from the heart; we are always praying for all the emperors: long life for them, a secure empire, a safe household, brave armies, a faithful senate, an upright people, a quiet world, and whatever else a man and a Caesar would wish.

In the defence against the charge of treason (Apology 30 to 33): Christians will not sacrifice to the emperors, but they pray to the true God for them, and swear by their safety rather than by their genii. Read from The Latin Library text, as elsewhere in this dossier.

Tertullian c. 155–c. 220 ADTo Scapula 2.6–7

A Christian is no one's enemy, much less the emperor's, whom he knows to be appointed by his God, and so must love, revere and honor him, and wish him safe with the whole Roman empire, as long as the world stands. ... So we honor the emperor in the way that is lawful for us and good for him: as a man second to God, who has received whatever he is from God, and is less only than God.

Written to the proconsul of Africa during persecution.