Monstrous surely is the madness of the human mind, that it is more disposed to charge God with unrighteousness than to blame itself for blindness.
Before answering positively, allow me to object to the scripture you cited as an argument to the contrary. The passage 1 Corinthians 10:13, cannot be interpreted to mean that God only commands men that which they are able to attain.
First, the passage does not refer to a command in the nature of your question, but rather to temptation. The distinction is important. A command originates in the holiness of God, while a temptation originates in the aberration of sin. However, to give you the benefit of the doubt, suppose that the passage did say that no command would be given except that which could be followed.
The audience to which the passage was written was not a universal one, and so you ought not apply universally. Paul addresses his letter thus,
To the church of God that is in Corinth, to those sanctified in Christ Jesus, called to be saints together with all those who in every place call upon the name of our Lord Jesus Christ, both their Lord and our:
Grace to you and peace from God our Father and the Lord Jesus Christ.
That is, the letter is written to the Church in Corinth, and the Body of Christ elsewhere. So, the “you” in the passage you cited cannot be generalized beyond that. Then, even if it says that God will not command “you” in a manner “you” cannot follow, “you” here can only mean the Church.
In the passage, Paul has just recounted to the Church, in light of his concern regarding wickedness among them (e.g, 3:4, 5:1, ), of the judgements upon the wicked in Israel, that his elect among them might be preserved, according to his promise to Abraham and for the redemption of the gentiles, among whom they were counted. And so in like manner the passage is an encouragement to the whole church, that as his people then were preserved among those trials while the wicked were purged from among them, God is still faithful to preserve the elect.
Now, the converse, is not as you suggest, that God is unjust to offer a command that the wicked cannot attain, but rather that God would be unfaithful to his promise if he were to allow his Church to be overcome. And since it cannot be said that God is in any sense unfaithful, we are assured of the perseverance of the Church.
Since I have addressed the passage, let me address the question you actually asked.
Does God, according to Calvinism, command people He has specifically given neither the ability nor choice to do so to repent and believe in Christ or be damned?
The call to repentance is universal. In his commentary on Acts 17:30, Calvin reiterates Paul’s teaching,
In these words Paul teacheth that we must give ear to God so soon as he speaketh, as it is written, “Today, if ye will hear his voice harden not your hearts,” ( Psalms 95:7; Hebrews 3:7.) For the stubbornness of those men is without excuse, who neglect this opportunity when God doth gently call them unto him. Also, we gather out of this place to what end the gospel is preached, to wit, that God may gather us to himself from the former errors of our life. Therefore, so oft as the voice of the gospel doth sound in our ears, let us know that God doth exhort us unto repentance.
And again he confirms the universal call to repentance in his commentary on Romans 3:23,
He urges on all, without exception, the necessity of seeking righteousness in Christ; as though he had said, “There is no other way of attaining righteousness; for some cannot be justified in this and others in that way; but all must alike be justified by faith, because all are sinners, and therefore have nothing for which they can glory before God.”
Now, since it is clear both that the call to repentance is universal, and also that not all repent, it is certain that God commands of many an act with which they cannot comply, by their will or ability. So, to answer your question in the affirmative, the command is made to all, and can be attained by none, so indeed God commands both the elect and the reprobate alike to repentance though they have neither the ability nor choice.
The nature of the question though, I conclude, is whether God makes the command, knowing full well that they are unable to comply- that is, you ask not “Does God do so?” because you already know the answer, but rather, “How does God do so justly?” I’ll continue after reminding you that one need not visit Calvin for an answer, but simply look to Paul’s answer to this very question in Roman’s 9.
What shall we say then? Is there injustice on God's part? By no means! For he says to Moses, “I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.” So then it depends not on human will or exertion, but on God, who has mercy. For the Scripture says to Pharaoh, “For this very purpose I have raised you up, that I might show my power in you, and that my name might be proclaimed in all the earth.” So then he has mercy on whomever he wills, and he hardens whomever he wills.
You will say to me then, “Why does he still find fault? For who can resist his will?” But who are you, O man, to answer back to God? Will what is molded say to its molder, “Why have you made me like this?” Has the potter no right over the clay, to make out of the same lump one vessel for honorable use and another for dishonorable use?
You might also consider Calvin’s discussion of your very question in “A Treatise of the Eternal Predestination of God,”
For, pretending a great concern for the honour of God, they bark at us, as imputing to Him a cruelty utterly foreign to His nature. Pighius denies that he has any contest with God. What cause, or whose cause is it, then, that Paul maintains? After he had adopted the. above axiom--that God hardens whom He will and has mercy on whom He will--he subjoins the supposed taunt of a wicked reasoner: " Why doth He yet find fault? For who hath resisted His will? " (Rom. ix. 19.) He meets such blasphemy as this by simply setting against it the power of God. If those clothe God with the garment of a tyrant, who refer the hardening of men even to His eternal counsel, we most certainly are not the originators of this doctrine. If they do God an injury who set His will above all other causes, Paul taught this doctrine long before us. Let these enemies of God, then, dispute the matter with the apostle. For I maintain nothing, in the present discussion, but what I declare is taught by him. About these barking dogs, however, I would not be very anxious.
So how can God condemn them if they are not free to choose?
The reprobate, though depraved, are not automatons. Though their will is not free because of their nature, the vice of the wicked is voluntary. In Book 2 of his Institutes, Calvin writes
Another objection [to predestination] is founded on a mode of speaking which is constantly observed both in Scripture and in common discourse. God works are said to be ours, and we are said to do what is holy and acceptable to God, just as we are said to commit sin. But if sins are justly imputed to us, as proceeding from ourselves, for the same reason (say they) some share must certainly be attributed to us in works of righteousness. It could not be accordant with reason to say, that we do those things which we are incapable of doing of our own motion, God moving us, as if we were stones. These expressions, therefore, it is said, indicate that while, in the matter of grace, we give the first place to God, a secondary place must be assigned to our agency.
In addition, their condemnation is not from their actions alone, but they are guilty of the very sin of Adam.
For whence that impotence of which the wicked so readily avail themselves as an excuse, but just because Adam voluntarily subjected himself to the tyranny of the devil? Hence the corruption by which we are held bound as with chains, originated in the first man’s revolt from his Maker. If all men are justly held guilty of this revolt, let them not think themselves excused by a necessity in which they see the clearest cause of their condemnation. […] The second step in the reasoning is vicious, because it leaps from voluntary to free; whereas we have proved above, that a thing may be done voluntarily, though not subject to free choice.
So ultimately, the premise of your question that men ought to be inculpable because of their depravity is circular, because their depravity is the result of Adam’s choice, which was truly free. And their own wills, through him depraved, are carried out in wickedness in their own bodies voluntarily. Their choice to sin is inseparable from Adam’s, which was free.
Again, Paul clearly explains the reason thusly: the default is condemnation, because God is just. And the reason that he spares some is plainly evident: to demonstrate his mercy. If we had the light of the Spirit, which by the Law uncovers our feebleness beside a holy God, we ought not think “Why is it that God condemns some, though they cannot in their own power repent?”, but to the contrary, “Why is it that God has redeemed some, though they are deserving of the same condemnation as the whole lot?”
Calvin concludes his discussion of the “difficulty” of the injustice of God,
In order to remove this difficulty, Paul divides his subject into two parts; in the, former of which he speaks of the elect, and in the latter of the reprobate; and in the one he would have us to contemplate the mercy of God, and in the other to acknowledge his righteous judgment. His first reply is, that the thought that there is injustice with God deserves to be abhorred, and then he shows that with regard to the two parties, there can be none.
But before we proceed further, we may observe that this very objection clearly proves, that inasmuch as God elects some and passes by others, the cause is not to be found in anything else but in his own purpose; for if the difference had been based on works, Paul would have to no purpose mentioned this question respecting the unrighteousness of God, no suspicion could have been entertained concerning it if God dealt with every one according to his merit. It may also, in the second place, be noticed, that though he saw that this doctrine could not be touched without exciting instant clamours and dreadful blasphemies, he yet freely and openly brought it forward; nay, he does not conceal how much occasion for murmuring and clamour is given to us, when we hear that before men are born their lot is assigned to each by the secret will of God; and yet, notwithstanding all this, he proceeds, and without any subterfuges, declares what he had learned from the Holy Spirit. It hence follows, that their fancies are by no means to be endured, who aim to appear wiser than the Holy Spirit, in removing and pacifying offences. That they may not criminate God, they ought honestly to confess that the salvation or the perdition of men depends on his free election. Were they to restrain their minds from unholy curiosity, and to bridle their tongues from immoderate liberty, their modesty and sobriety would be deserving of approbation; but to put a restraint on the Holy Spirit and on Paul, what audacity it is! Let then such magnanimity ever prevail in the Church of God, as that godly teachers may not be ashamed to make an honest profession of the true doctrine, however hated it may be, and also to refute whatever calumnies the ungodly may bring forward.